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+arXiv:2301.00266v1  [math.SG]  31 Dec 2022
+Action-angle coordinates and KAM theory for
+singular symplectic manifolds
+Eva Miranda
+Arnau Planas
+Laboratory of Geometry and Dynamical Systems, Department of
+Mathematics & IMTech, Universitat Polit`ecnica de Catalunya, Barcelona
+and CRM, Centre de Recerca Matem`atica, Bellaterra
+Current address: UPC-Edifici P, Avinguda del Doctor Maran´on, 44-50, 08028,
+Barcelona, Spain
+Email address: evamiranda@upc.edu
+Department of Mathematics, Universitat Polit`ecnica de Catalunya,
+Barcelona
+Email address: arnau.planas.bahi@gmail.com
+
+2020 Mathematics Subject Classification.
+53D05, 53D20, 70H08, 37J35, 37J40 ;
+37J39, 58D19
+Key words and phrases. amsbook, AMS-LATEX
+Eva Miranda is supported by the Catalan Institution for Research and Advanced
+Studies via an ICREA Academia Prize 2016 and ICREA Academia Prize 2021.
+Eva Miranda is also supported by the Spanish State Research Agency, through the
+Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence
+in R&D (project CEX2020-001084-M). Eva Miranda also acknowledges partial
+support from the grant “Computational, dynamical and geometrical complexity in
+fluid dynamics”, Ayudas Fundaci´on BBVA a Proyectos de Investigaci´on Cient´ıfica
+2021. Both authors are supported by the project PID2019-103849GB-I00 of the
+Spanish State Agency AEI /10.13039/501100011033.
+Dedicated to the memory of Amelia Galcer´an Sorribes.
+
+Contents
+Preface
+vii
+Part 1.
+Introduction and preliminaries
+1
+Chapter 1.
+Introduction
+3
+1.1.
+Structure and results of this monograph
+3
+Chapter 2.
+A primer on singular symplectic manifolds
+7
+2.1.
+b-Poisson manifolds
+7
+2.2.
+On bm-Symplectic manifolds
+10
+2.3.
+Desingularizing bm-Poisson manifolds
+12
+Chapter 3.
+A crash course on KAM theory
+15
+Part 2.
+Action-angle coordinates and cotangent models
+19
+Chapter 4.
+An action-angle theorem for bm-symplectic manifolds
+23
+4.1.
+Basic definitions
+23
+4.2.
+On bm-integrable systems
+24
+4.3.
+Examples of bm-integrable systems
+26
+4.4.
+Looking for a toric action
+29
+4.5.
+Action-angle coordinates on bm-symplectic
+manifolds
+32
+Chapter 5.
+Reformulating the action-angle coordinate via cotangent lifts
+37
+5.1.
+Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry
+37
+5.2.
+The case of bm-symplectic manifolds
+38
+Part 3.
+A KAM theorem for bm-symplectic manifolds
+41
+Chapter 6.
+A new KAM theorem
+43
+6.1.
+On the structure of the proof
+43
+6.2.
+Technical results
+50
+6.3.
+A KAM theorem on bm-symplectic manifolds
+69
+Chapter 7.
+Desingularization of bm-integrable systems
+93
+Chapter 8.
+Desingularization of the KAM theorem on bm-symplectic
+manifolds
+97
+Chapter 9.
+Potential applications to Celestial mechanics
+101
+9.1.
+The Kepler Problem
+101
+v
+
+vi
+CONTENTS
+9.2.
+The Problem of Two Fixed Centers
+102
+9.3.
+Double Collision and McGehee coordinates
+103
+9.4.
+The restricted three-body problem
+105
+Bibliography
+107
+
+Preface
+I confess I envy the planets —
+they’ve got their own orbits and
+nothing stands on their way.
+Intermezzo,
+Mykhailo
+Kotsi-
+ubynsky.
+This monograph explores classification and perturbation problems for inte-
+grable systems on a class of Poisson manifolds called bm-Poisson manifolds. This
+is the first class of Poisson manifolds for which perturbation theory is established
+outside the symplectic category. Even if the class of bm-Poisson manifolds is not
+ample enough to represent the wild set of Poisson manifolds, this investigation can
+be seen as a first step for the study of perturbation theory for general Poisson man-
+ifolds. In view of the work of the first author with Nest, the theorems established
+in this monograph constitute more than a mild generalization in Poisson Geometry
+and, this toy example, sets the path to consider KAM theory in the general realm
+of Poisson manifolds. Reduction theorems and bm-symplectic manifolds have been
+recently explored in [MM22]. This monograph contributes to the theory opening
+the investigation of perturbation theory on these manifolds thus completing other
+facets in the study of their dynamics as the recent work on the Arnold conjecture
+[BMO22].
+Symplectic geometry has been the common language of physics as the position-
+momentum tandem can be modelled over a cotangent bundle. Cotangent bundles
+are naturally endowed with a symplectic form which is a non-degenerate closed
+2-form. The symplectic form of the cotangent bundle is given by the differential of
+the Liouville one-form.
+bm-Poisson manifolds are manifolds that are symplectic away from a hyper-
+surface along which they satisfy some transversality properties. They often model
+problems on symplectic manifolds with boundary such as the study of their de-
+formation quantization and celestial mechanics. As on the complementary of the
+critical set the manifolds are symplectic, extending the investigation of Hamiltonian
+dynamics to this realm is key to understand Hamiltonian Dynamics on compact-
+ification of symplectic manifolds. Several regularization transformations used in
+celestial mechanics (as McGehee or Moser regularization) provide examples of such
+compactifications.
+One of the interesting properties of bm-Poisson manifolds is that their investi-
+gation can be achieved considering the language of bm-forms. That is to say, we can
+work with forms that are symplectic away from the critical set and admit a smooth
+extension as a form over a Lie algebroid generalizing De Rham forms as form over
+the standard Lie algebroid of the tangent bundle of the manifold.
+To consider
+bm-forms the standard tangent bundle is replaced by the bm-tangent bundle. This
+vii
+
+viii
+PREFACE
+allows us to mimic symplectic geometry by replacing the cotangent bundle by the
+dual of the bm-tangent bundle. However, Poisson geometry leaves its footprint and
+new invariants which can be identified as the modular class of the Poisson structure
+arise already at the semilocal level.
+Contrary to the initial expectations, several of the results for bm-symplectic
+manifolds do not resemble the b-case so far. Considering these more general singu-
+larities yields a better understanding of the general Poisson case and the different
+levels of complexity. As an illustration of this phenomena: in the study of quan-
+tization of those systems an interesting pattern makes the quantization radically
+different in the even and odd case [GMW18b, GMW21] and the resulting model
+is finite-dimensional in the b-case. Understanding how the different degrees m are
+related is a hard task: The desingularization technique introduced by Guillemin-
+Miranda-Weitsman in [GMW17] turned out to have important applications in the
+investigation of complexity properties of toric bm-symplectic manifolds [GMW18a]
+and to the study of the Arnold conjecture in this set-up [BMO22]. In this mono-
+graph we explore a new facet of these manifolds: that of perturbation theory.
+In the second part of the monograph we consider integrable systems on these
+manifolds turn out to have associated generalized Hamiltonian actions of tori in
+a neighbourhood of a Liouville torus. We use this generalized Hamiltonian group
+action to prove existence of action-angle coordinates in a neighborhood of a Liouville
+torus. The action-angle coordinate theorem that we prove gives a semilocal normal
+form in the neighbourhood of a Liouville torus for the bm-symplectic structure
+which depends on the modular weight of the connected component of the critical
+set in which the Liouville torus is lying and the modular weights of the associated
+toric action. This action-angle theorem allows us to identify a neighborhood of
+the Liouville torus with the bm-cotangent lift of the action of a torus acting by
+translations on itself. This interpretation of the action-angle theorem as cotangent
+lift allows us to identify the modular weight as their only semilocal invariant. In
+doing so, we compare this action-angle coordinate theorem with the classical action-
+angle coordinate theorems for symplectic manifolds and an action-angle theorem
+for folded symplectic manifolds ([CM22]).
+In part 3 of the monograph we study perturbation theory in this new set-up
+and examine some potential applications to physical systems. In particular, we
+prove a KAM theorem for bm-Poisson manifolds which clearly refines and improves
+the one obtained for b-Poisson manifolds in [KMS16a]. As an outcome of this
+result together with the extension of the desingularization techniques of Guillemin-
+Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem
+for folded symplectic manifolds where KAM theory has never been considered be-
+fore. In the way, we also obtain a brand-new KAM theorem for symplectic manifolds
+where the perturbation keeps track of a distinguished hypersurface. In celestial me-
+chanics, this distinguished hypersurface can be the line at infinity or the collision
+set.
+Barcelona, December 2022, Eva Miranda and Arnau Planas
+
+Part 1
+Introduction and preliminaries
+
+
+CHAPTER 1
+Introduction
+Both symplectic and Poisson geometry emerge from the study of classical me-
+chanics. Both are broad fields widely studied and with powerful results. But as
+Poisson structures are far more general than the symplectic ones, most outstanding
+results in symplectic geometry do not translate well to Poisson manifolds. Here is
+where bm-Poisson structures come to play. bm-Poisson structures (or bm-symplectic
+structures) lie somewhere between these two worlds. They extend symplectic struc-
+tures but in a really controlled way. This is why fundamental results in symplectic
+geometry still work in bm-symplectic geometry. However, an adaptation of these
+theories like deformation or Moser theory requires some work (see [GMPS15a]
+and others).
+The study of bm-Poisson geometry sparked from the study of symplectic man-
+ifold with boundary [Mel93a]. In the last years the interest in this field increased
+after the classification result for b-Poisson structures obtained in [Rad02]. Later
+on, [GMP14] translated these structures to the language of forms and started
+applying symplectic tools to study them. A lot of papers in the following years
+studied different aspects of these structures: [GMP10], [GMP14], [GMPS15b],
+[GMW17], [MOT14] and [GLPR17] are some examples.
+Inspired by the study of manifolds with boundary, we work on a pair of mani-
+folds (M, Z) where Z is an hypersurface and call this pair b-manifold
+In this context, [Sco16] generalized the b-symplectic forms by allowing higher
+degrees of degeneracy of the Poisson structures.
+The bm-symplectic structures
+inherit most of the properties of b-symplectic structures. This booklet focuses on
+different aspects of the investigation of bm-symplectic structures covering mainly
+integrable systems and KAM theory. First, we present some preliminary notions
+necessary to address the problem of perturbation.
+We present an action-angle
+theorem for bm-Poisson structures and state and prove the KAM theory equivalent
+in manifolds with bm-symplectic structures.
+1.1. Structure and results of this monograph
+1.1.1. Part 1: Introduction and Preliminaries. In the preliminaries, we
+give the basic notions that lead to the questions we are addressing in this booklet.
+In the first part, we introduce the concept of b-Poisson manifolds or b-symplectic
+manifolds, a class of Poisson manifold which is symplectic outside a critical hyper-
+surface. It study comes motivated by the investigation of manifolds with boundary.
+Next, we talk about a generalization of these structures, that allow a higher degree
+of degeneracy of the structure: the bm-symplectic structures. These structures are
+the main focus of our investigations. A key concept that will play an important
+work in this book is the study of the desingularization of these singular structures.
+3
+
+4
+1. INTRODUCTION
+Finally, we give a short introduction to KAM theory, a theory that will be gener-
+alized in the setting of bm-manifolds in the last chapter.
+Motivation comes from several examples of singular symplectic structures ap-
+pearing naturally in classical problems of celestial mechanics which are discussed
+on the last chapter of the monograph. We also describe the difficulties of finding
+these examples, and the subtleties of dealing with these singular structures in the
+exploration of conservative systems.
+1.1.2. Part 2: Action-angle coordinates and cotangent models for bm-
+integrable systems. In this Chapter we define the concept of bm-functions and
+bm-integrable systems. We present several examples of bm-integrable systems that
+come from classical mechanics. After all this we present a version of the action-angle
+theorem for bm-symplectic manifolds.
+Theorem A. Let (M, x, ω, F) be a bm-integrable system, where F = (f1 =
+a0 log(x)+�m−1
+j=1 aj 1
+xj , f2, . . . , fn). Let m ∈ Z be a regular point, and such that the
+integral manifold through m is compact. Let Fm be the Liouville torus through m.
+Then, there exists a neighborhood U of Fm and coordinates (θ1, . . . , θn, σ1, . . . , σn) :
+U → Tn × Bn such that:
+(1) We can find an equivalent integrable system F = (f1 = a′
+0 log(x) +
+�m−1
+j=1 a′
+j
+1
+xj ) such that a′
+0, . . . , a′
+m−1 ∈ R,
+(2)
+ω|U =
+
+
+m
+�
+j=1
+c′
+j
+c
+σj
+1
+dσ1 ∧ dθn
+
+ +
+n
+�
+i=2
+dσi ∧ dθi
+where c is the modular period and c′
+j = −(j − 1)a′
+j−1, also
+(3) the coordinates σ1, . . . , σn depend only on fn, . . . fn.
+1.1.3. Part 3: KAM theory on bm-symplectic manifolds and applica-
+tions to Celestial Mechanics. In this chapter we provide several KAM theorem
+for (singular) symplectic manifolds including bm-symplectic manifolds.
+We begin by considering perturbation theory for bm-symplectic manifolds. Then
+we give an outline of how to construct the bm-symplectomorphism that will be the
+main character of the proof of the KAM theorem for bm-symplectic manifolds. After
+this, we show some technical results that are needed for the proof. These technical
+results even if quite similar to the standard KAM equivalents, have some subtleties
+that need to be addressed. We end the chapter with the proof of the bm-KAM
+theorem and several applications to establish KAM theorems in other singular sit-
+uations (folded symplectic manifolds) and on symplectic manifolds with prescribed
+invariant hypersurfaces.
+The first KAM theorem is the following:
+Theorem B. Let G ⊂ Rn, n ≥ 2 be a compact set. Let H(φ, I) = ˆh(I) +
+f(φ, I), where ˆh is a bm-function ˆh(I) = h(I) + q0 log(I1) + �m−1
+i=1
+qi
+Ii
+1 defined on
+Dρ(G), with h(I) and f(φ, I) analytic. Let ˆu = ∂ˆh
+∂I and u = ∂h
+∂I . Assume | ∂u
+∂I |G,ρ2 ≤
+M, |u|ξ ≤ L. Assume that u is µ non-degenerate (| ∂u
+∂I | ≥ µ|v| for some µ ∈ R+ and
+I ∈ G. Take a = 16M. Assume that u is one-to-one on G and its range F = u(G)
+is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1. Let
+
+1.1. STRUCTURE AND RESULTS OF THIS MONOGRAPH
+5
+(1)
+(1.1)
+ε := ∥f∥G,ρ ≤
+ν2µ2ˆρ2τ+2
+24τ+32L6M 3 γ2,
+(2)
+(1.2)
+γ ≤ min(8LMρ2
+ν ˆρτ+1 , L
+K′ )
+(3)
+(1.3)
+µ ≤ min(2τ+5L2M, 27ρ1L4Kτ+1, βντ+122τ+1ρτ
+1),
+where ˆρ := min
+�
+νρ1
+12(τ+2), 1
+�
+. Define the set ˆG = ˆGγ := {I ∈ G− 2γ
+µ |u(I) is τ, γ, c, ˆq−
+Dioph.}. Then, there exists a real continuous map T : W ρ1
+4 (Tn) × ˆG → Dρ(G) an-
+alytic with respect the angular variables such that
+(1) For all I ∈ ˆG the set T (Tn ×{I}) is an invariant torus of H, its frequency
+vector is equal to u(I).
+(2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)) with estimates
+|Tφ(φ, I)| ≤ 22τ+15ML2
+ν2ˆρ2τ+1
+ε
+γ2
+|TI(φ, I))| ≤ 210+τL(1 + M)
+ν ˆρτ+1
+ε
+γ
+(3) meas[(Tn ×G)\T (Tn× ˆG)] ≤ Cγ where C is a really complicated constant
+depending on n, µ, D, diamF, M, τ, ρ1, ρ2, K and L.
+Also, we obtain a way to associate a standard symplectic integrable system or
+a folded integrable system to a bm-integrable system, depending on the parity of m.
+This is done in such a way that the dynamics of the desingularized system are the
+same than the dynamics of the original one. So it defines a honest desingularization
+of the integrable system.
+Theorem C. The desingularization transforms a bm-integrable system into an
+integrable system on a symplectic manifold for even m.
+For m odd, the desin-
+gularization associates to it a folded integrable system.
+The integrable systems
+satisfy:
+Xω
+fj = Xωǫ
+fjǫ.
+This allows us to obtain two new KAM theorems using this desingularization
+combined with the former bm-KAM theorem. The first of these theorems is a KAM
+theorem for standard symplectic manifolds, where the perturbation has a particu-
+lar expression. This result is more restrictive than the standard KAM theorem but
+allow us to guarantee that the perturbations leave a given hypersurface invariant.
+This means that the tori belonging to that hypersurface remain on the hypersur-
+face after the perturbation. This can be interesting for a number of reasons and
+situations such as problems in Celestial mechanics where it is convenient to keep
+track of a particular hypersurface such as the line at infinity. The higher order
+singularities allow to consider perturbations that are tangent to the hypersurface
+up to a certain order.
+
+6
+1. INTRODUCTION
+Theorem D. Consider a neighborhood of a Liouville torus of an integrable
+system Fε as in 8.1 of a symplectic manifold (M, ωε) semilocally endowed with
+coordinates (I, φ), where φ are the angular coordinates of the torus, with ωε =
+c′dI1 ∧dφi +�n
+j=1 dIj ∧dφj. Let H = (m−1)cm−1c′I1 +h(˜I)+R(˜I, ˜φ) be a nearly
+integrable system where
+�
+˜I1
+=
+c′ Im+1
+1
+m+1 ,
+˜φ1
+=
+c′Im
+1 φ1,
+and
+� ˜I
+=
+(˜I1, I2, . . . , In),
+˜φ
+=
+(˜φ1, φ2, . . . , φn).
+Then the results for the bm-KAM theorem 6.3 applied to Hsing =
+1
+I2k−1
+1
++ h(I) +
+R(I, φ) hold also for this desingularized system.
+The second one is a KAM theorem for folded-symplectic manifolds, where KAM
+theory has not been considered to-date.
+Theorem E. Consider a neighborhood of a Liouville torus of an integrable
+system Fε as in 8.2 of a folded symplectic manifold (M, ωε) semilocally endowed
+with coordinates (I, φ), where φ are the angular coordinates of the Torus, with
+ωε = 2cI1dI1 ∧ dφ1 + �m
+j=2 dIj ∧ dφj. Let H = (m − 1)cm−1cI2
+1 + h(˜I) + R(˜I, ˜φ) a
+nearly integrable system with�
+˜I1
+=
+2c Im+2
+1
+m+2 ,
+˜φ1
+=
+2cIm+1
+1
+φ1,
+and
+� ˜I
+=
+(˜I1, I2, . . . , In),
+˜φ
+=
+(˜φ1, φ2, . . . , φn).
+Then the results for the bm-KAM theorem 6.3 applied to Hsing =
+1
+I2k
+1 +h(I)+R(I, φ)
+also hold for this desingularized system.
+Last but not least, we illustrate the connection between bm-symplectic struc-
+tures and classical mechanics by providing several examples.
+Several potential
+applications to celestial mechanics are discussed.
+
+CHAPTER 2
+A primer on singular symplectic manifolds
+In this first chapter of the booklet we introduce basic notions on singular sym-
+plectic structures, as well as some concepts on standard KAM theory. Those are
+the two main pillars of this monograph.
+Let M be a smooth manifold, a Poisson structure on M is a bilinear map
+{·, ·} : C∞(M) × C∞(M) → C∞(M) which is skew-symmetric and satisfies both
+the Jacobi identity and the Leibniz rule. It is possible to express {f, g} in terms
+of a bivector field via the following equality {f, g} = Π(df ∧ dg) with Π a section
+of Λ2(T M). Π is the associated Poisson bivector. We will use indistinctively
+the terminology of Poisson structure when referring to the bracket or the Poisson
+bivector.
+A b-Poisson bivector field on a manifold M 2n is a Poisson bivector such that
+the map
+(2.1)
+F : M →
+2n
+�
+T M : p �→ (Π(p))n
+is transverse to the zero section. Then, a pair (M, Π) is called a b-Poisson man-
+ifold and the vanishing set Z of F is called the critical hypersurface. Observe
+that Z is an embedded hypersurface.
+This class of Poisson structures was studied by Radko [Rad02] in dimension
+two and considered in numerous papers in the last years: [GMP10], [GMP14],
+[GMPS15b], [GMW17], [MOT14] and [GLPR17] among others.
+2.1. b-Poisson manifolds
+Next, we recall classification theorem of b-Poisson surfaces as presented by Olga
+Radko and the cohomological re-statement and proof given by Guillemin, Miranda
+and Pires in [GMP14].
+In what follows, (M, Π) will be a closed smooth surface with a b-Poisson struc-
+ture on it, and Z its critical hypersurface.
+Let h be the distance function to Z as in [MOT14]1.
+Definition 2.1. The Liouville volume of (M, Π) is the following limit:
+V (Π) := limǫ→0
+�
+|h|>ǫ ωn2.
+The previous limit exists and it is independent of the choice of the defining
+function h of Z (see [Rad02] for the proof).
+Definition 2.2. For any (M, Π) oriented Poisson manifold, let Ω be a volume
+form on it, and let uf denote the Hamiltonian vector field of a smooth function
+1Notice the difference with [Rad02] where h is assumed to be a global defining function.
+2For surfaces n = 1.
+7
+
+8
+2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS
+f : M → R. The modular vector field XΩ is the derivation defined as follows:
+f �→ Luf Ω
+Ω
+.
+Definition 2.3. Given γ a connected component of the critical set Z(Π) of a
+closed b-Poisson manifold (M, Π), the modular period of Π around γ is defined
+as:
+Tγ(Π) := period of XΩ|γ.
+Remark 2.4. The modular vector field XΩ of the b-Poisson manifold (M, Z)
+does not depend at Z on the choice of Ω because for different choices for volume
+form the difference of modular vector fields is a Hamiltonian vector field. Observe
+that this Hamiltonian vector field vanishes on the critical set as Π vanishes there
+too.
+Definition 2.5. Let Mn(M) = Cn(M)/ ∼ where Cn(M) is the space of dis-
+joint oriented curves and ∼ identifies two sets of curves if there is an orientation-
+preserving diffeomorphism mapping the first one to the second one and preserving
+the orientations of the curves.
+The following theorem classifies b-symplectic structures on surfaces using these
+invariants:
+Theorem 2.6 (Radko [Rad02]). Consider two b-Poisson structures Π, Π′ on
+a closed orientable surface M. Denote its critical hypersurfaces by Z and Z′. These
+two b-Poisson structures are globally equivalent (there exists a global orientation
+preserving diffeomorphism sending Π to Π′) if and only if the following coincide:
+• the equivalence classes of [Z] and [Z′] ∈ Mn(M),
+• their modular periods around the connected components of Z and Z′,
+• their Liouville volume.
+An appropriate formalism to deal with these structures was introduced in
+[GMP10].
+Definition 2.7. A b-manifold3 is a pair (M, Z) of a manifold and an embed-
+ded hypersurface.
+In this way, the concept of b-manifold previously introduced by Melrose is
+generalized to consider additional geometric structures on the manifold.
+Definition 2.8. A b-vector field on a b-manifold (M, Z) is a vector field
+tangent to the hypersurface Z at every point p ∈ Z.
+Definition 2.9. A b-map from (M, Z) to (M ′, Z′) is a smooth map φ : M →
+M ′ such that φ−1(Z′) = Z and φ is transverse to Z′.
+Observe that if x is a local defining function for Z and (x, x1, . . . , xn−1) are
+local coordinates in a neighborhood of p ∈ Z then the C∞(M)-module of b-vector
+fields has the following local basis
+(2.2)
+{x ∂
+∂x, ∂
+∂x1
+, . . . ,
+∂
+∂xn−1
+}.
+3The ‘b’ of b-manifolds stands for ‘boundary’, as initially considered by Melrose (Chapter 2
+of [Mel93b]) for the study of pseudo-differential operators on manifolds with boundary.
+
+2.1. b-POISSON MANIFOLDS
+9
+Figure 1. Artistic representation of a b-function on a b-manifold
+near the critical hypersurface.
+In contrast to [GMP10], in this monograph we are not requiring the existence
+of a global defining function for Z and orientability of M. However, we require the
+existence of a defining function in a neighborhood of each point of Z. By relaxing
+this condition, the normal bundle of Z need not be trivial.
+Given (M, Z) a b-manifold, [GMP10] shows that there exists a vector bundle,
+denoted by bT M whose smooth sections are b-vector fields. This bundle is called
+the b-tangent bundle of (M, Z).
+The b-cotangent bundle bT ∗M is defined using duality. A b-form is a section
+of the b-cotangent bundle. Around a point p ∈ Z the C∞(M)-module of these
+sections has the following local basis:
+(2.3)
+{ 1
+xdx, dx1, . . . , dxn−1}.
+In the same way we define a b-form of degree k to be a section of the bundle
+�k(bT ∗M), the set of these forms is denoted bΩk(M). Denoting by f the distance
+function4 to the critical hypersurface Z, we may write the following decomposition
+as in [GMP10] for any ω ∈b Ωk(M) :
+(2.4)
+ω = α ∧ df
+f + β, with α ∈ Ωk−1(M) and β ∈ Ωk(M).
+This decomposition allows to extend the differential of the de Rham complex
+d to bΩ(M) by setting dω = dα ∧ df
+f + dβ.
+Degree 0 functions are called b-functions and and near Z can be written as
+c log |x| + g,
+where c ∈ R, g ∈ C∞, and x is a local defining function.
+The associated cohomology is called b-cohomology and it is denoted by bH∗(M).
+Definition 2.10. A b-symplectic form on a b-manifold (M 2n, Z) is defined
+as a non-degenerate closed b-form of degree 2 (i.e., ωp is of maximal rank as an
+element of Λ2( bT ∗
+p M) for all p ∈ M).
+The notion of b-symplectic forms is dual to the notion of b-Poisson structures.
+The advantage of using forms rather than bivector fields is that symplectic tools
+can be ‘easily’ exported.
+4Originally in [GMP10] f stands for a global function, but for non-orientable manifolds we
+may use the distance function instead.
+
+10
+2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS
+Radko’s classification theorem [Rad02] can be translated into this language.
+This translation was already formulated in [GMP10]:
+Theorem 2.11 (Radko’s theorem in b-cohomological language, [GMP14]).
+Let S be a closed orientable surface and let ω0 and ω1 be two b-symplectic forms on
+(S, Z) defining the same b-cohomology class (i.e.,[ω0] = [ω1]). Then there exists a
+diffeomorphism φ : S → S such that φ∗ω1 = ω0.
+2.2. On bm-Symplectic manifolds
+2.2.1. Basic definitions. By relaxing the transversality condition allowing
+higher order singularities ([Arn89] and [AA81]) we may consider other symplectic
+structures with singularities as done by Scott [Sco16] with bm-symplectic struc-
+tures.
+Let m be a positive integer a bm-manifold is a b-manifold (M, Z) together
+with a bm-tangent bundle attached to it. The bm-tangent bundle is (by Serre-Swan
+theorem [Swa62]) a vector bundle, bmT M whose sections are given by,
+Γ(bmT M) = {v ∈ Γ(T M) : v(x)
+vanishes to order m at Z},
+where x is a defining function for the critical set Z in a neighborhood of each
+connected component of Z and can be defined as x : M \ Z → (0, ∞), x ∈ C∞(M)
+such that:
+• x(p) = d(p) a distance function from p to Z for p : d(p) ≤ 1/2
+• x(p) = 1 on M \ {p ∈ M such that d(p) < 1}.5
+(This definition of x allows us to extend the construction in [Sco16] to the non-
+orientable case as in [MOT14].) We may define the notion of a bm-map as a map
+in this category (see [Sco16]).
+The sections of this bundle are referred to as bm-vector fields and their flows
+define bm-maps. In local coordinates, the sections of the bm-tangent bundle are
+generated by:
+(2.5)
+{xm ∂
+∂x, ∂
+∂x1
+, . . . ,
+∂
+∂xn−1
+}.
+Proceeding mutatis mutandis as in the b-case one defines the bm-cotangent
+bundle (bmT ∗M), the bm-de Rham complex and the bm-symplectic structures.
+A Laurent Series of a closed bm-form ω is a decomposition of ω in a tubular
+neighborhood U of Z of the form
+(2.6)
+ω = dx
+xm ∧ (
+m−1
+�
+i=0
+π∗(αi)xi) + β
+with π : U → Z the projection of the tubular neighborhood onto Z, αi a closed
+smooth de Rham form on Z and β a de Rham form on M.
+In [Sco16] it is proved that in a neighborhood of Z, every closed bm-form ω
+can be written in a Laurent form of type (2.6) having fixed a (semi)local defining
+function.
+bm-Cohomology is related to de Rham cohomology via the following theorem:
+5Then a bm-manifold will be a triple (M, Z, x), but for the sake of simplicity we refer to it
+as a pair (M, Z) and we tacitly assume that the function x is fixed.
+
+2.2. ON bm-SYMPLECTIC MANIFOLDS
+11
+Theorem 2.12 (bm-Mazzeo-Melrose, [Sco16]). Let (M, Z) be a bm-manifold,
+then:
+(2.7)
+bmHp(M) ∼= Hp(M) ⊕ (Hp−1(Z))m.
+The isomorphism constructed in the proof of the theorem above is non-canonical
+(see [Sco16]).
+The Moser path method can be generalized to bm-symplectic structures (see
+[MS21] for the generalization from surfaces in [Sco16] to general manifolds):
+Theorem 2.13 (Moser path method). Let ωt be a path of bm-symplectic
+forms defining the same bm-cohomology class [ωt] on (M 2n, Z) with M 2n closed
+and orientable then there exist a bm-symplectomorphism ϕ : (M 2n, Z) −→ (M 2n, Z)
+such that ϕ∗(ω1) = ω0.
+An outstanding consequence of Moser path method is a global classification
+of closed orientable bm-symplectic surfaces `a la Radko in terms of bm-cohomology
+classes.
+Theorem 2.14 (Classification of closed orientable bm-surfaces, [Sco16]).
+Let ω0 and ω1 be two bm-symplectic forms on a closed orientable connected bm-
+surface (S, Z). Then, the following conditions are equivalent:
+• their bm-cohomology classes coincide [ω0] = [ω1],
+• the surfaces are globally bm-symplectomorphic,
+• the Liouville volumes of ω0 and ω1 and the numbers
+�
+γ
+αi
+for all connected components γ ⊆ Z and all 1 ≤ i ≤ m coincide (where
+αi are the one-forms appearing in the Laurent decomposition of the two
+bm-forms of degree 2, ω0 and ω1).
+Definition 2.15. The numbers [αi] =
+�
+γ αi are called modular weights for the
+connected components γ ⊂ Z.
+A relative version of Moser’s path method is proved in [GMW17]. As a corol-
+lary we obtain the following local description of a bm-symplectic manifold:
+Theorem 2.16 (bm-Darboux theorem, [GMW17]). Let ω be a bm-symplectic
+form on (M, Z) and p ∈ Z. Then we can find a coordinate chart (U, x1, y1, . . . , xn, yn)
+centered at p such that on U the hypersurface Z is locally defined by x1 = 0 and
+ω = dx1
+xm
+1
+∧ dy1 +
+n
+�
+i=2
+dxi ∧ dyi.
+Remark 2.17. For the sake of simplicity sometimes we will omit any explicit
+reference to the critical set Z and we will talk directly about bm-symplectic struc-
+tures on manifolds M implicitly assuming that Z is the vanishing locus of Πn where
+Π is the Poisson vector field dual to the bm-symplectic form.
+Next, we present two lemmas that allow us to talk about bm-symplectic struc-
+tures and bm-Poisson as two different presentations of the same geometrical struc-
+ture on a b-manifold. The lemma below shows that they are dual to each other
+and, thus, in one-to-one correspondence.
+
+12
+2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS
+Lemma 2.18. Let ω be a bm-symplectic and Π its dual vector field, then Π is a
+bm-Poisson structure.
+Proof. The quickest way to do this is to take the inverse, which is a bivector
+field, and observe that it is a Poisson structure (because dω = 0 implies [Π, Π] = 0).
+To see that it is bm-Poisson it is enough to check it locally for any point along the
+critical set. Take a point p on the critical set Z and apply the bm-Darboux theorem
+to get ω = dx1/xm
+1 ∧ dy1 + �
+i>1 dxi ∧ dyi This means that in the new coordinate
+system
+Π = xm
+1
+∂
+∂x1
+∧
+∂
+∂y1
++
+�
+i>1
+∂
+∂xi
+∧ ∂
+∂yi
+and thus Π is a bm-Poisson structure.
+□
+Conversely,
+Lemma 2.19. Let Π be bm-Poisson and ω its dual vector field, then ω is a
+bm-symplectic structure.
+Proof. If Π transverse `a la Thom on Z with singularity of order m then
+because of Weinstein’s splitting theorem we can locally write
+Π = xm
+1
+∂
+∂x1
+∧
+∂
+∂y1
++
+�
+i>1
+∂
+∂xi
+∧ ∂
+∂yi
+now its inverse is ω = dx1/xm
+1 ∧ dy1 + �
+i>1 dxi ∧ dyi which is a bm-symplectic
+form.
+□
+Hence we have a correspondence from bm-symplectic structures to bm-Poisson
+structures.
+2.3. Desingularizing bm-Poisson manifolds
+In [GMW17] Guillemin, Miranda and Weitsman presented a desingularization
+procedure for bm-symplectic manifolds proving that we may associate a family of
+folded symplectic or symplectic forms to a given bm-symplectic structure depending
+on the parity of m. Namely,
+Theorem 2.20 (Guillemin-Miranda-Weitsman, [GMW17]). Let ω be a
+bm-symplectic structure on a closed orientable manifold M and let Z be its critical
+hypersurface.
+• If m = 2k, there exists a family of symplectic forms ωǫ which coincide with
+the bm-symplectic form ω outside an ǫ-neighborhood of Z and for which
+the family of bivector fields (ωǫ)−1 converges in the C2k−1-topology to the
+Poisson structure ω−1 as ǫ → 0 .
+• If m = 2k + 1, there exists a family of folded symplectic forms ωǫ which
+coincide with the bm-symplectic form ω outside an ǫ-neighborhood of Z.
+As a consequence of Theorem 2.20, any closed orientable manifold that supports
+a b2k-symplectic structure necessarily supports a symplectic structure.
+In [GMW17] explicit formulae are given for even and odd cases. Let us refer
+here to the even-dimensional case as these formulae will be used later on.
+
+2.3. DESINGULARIZING BM-POISSON MANIFOLDS
+13
+Let us briefly recall how the desingularization is defined and the main result in
+[GMW17]. Recall that we can express the b2k-form as:
+(2.8)
+ω = dx
+x2k ∧
+�2k−1
+�
+i=0
+xiαi
+�
++ β.
+This expression holds on a ǫ-tubular neighborhood of a given connected com-
+ponent of Z. This expression comes directly from equation 2.6, to see a proof of
+this result we refer to [Sco16].
+Definition 2.21. Let (S, Z, x), be a b2k-manifold, where S is a closed orientable
+manifold and let ω be a b2k-symplectic form. Consider the decomposition given by
+the expression (2.8) on an ǫ-tubular neighborhood Uǫ of a connected component of
+Z.
+Let f ∈ C∞(R) be an odd smooth function satisfying f ′(x) > 0 for all x ∈ [−1, 1]
+and satisfying outside that
+(2.9)
+f(x) =
+�
+−1
+(2k−1)x2k−1 − 2
+for
+x < −1,
+−1
+(2k−1)x2k−1 + 2
+for
+x > 1.
+Let fǫ(x) be defined as ǫ−(2k−1)f(x/ǫ).
+The fǫ-desingularization ωǫ is a form that is defined on Uǫ by the following
+expression:
+ωǫ = dfǫ ∧
+�2k−1
+�
+i=0
+xiαi
+�
++ β.
+This desingularization procedure is also known as deblogging in the literature.
+Remark 2.22. Though there are infinitely many choices for f, we will assume
+that we choose one, and assume it fixed through the rest of the discussion.
+It
+would be interesting to discuss the existence of an isotopy of forms under a change
+of function f.
+Remark 2.23. Because ωǫ can be trivially extended to the whole S in such a
+way that it agrees with ω (see [GMW17]) outside a neighborhood of Z, we can
+talk about the fǫ-desingularization of ω as a form on S.
+
+
+CHAPTER 3
+A crash course on KAM theory
+The last part of this monograph is entirely dedicated to prove a KAM theorem
+for bm-symplectic structures and to find applications. So the aim of this section is
+to give a quick overview of the traditional KAM theorem. The setting of the KAM
+theorem is a symplectic manifold with action-angle coordinates and an integrable
+system in it. The theorem says that under small perturbations of the Hamiltonian
+”most” of the Liouville tori survive.
+Consider Tn×G ⊂ Tn×Rn with action-angle coordinates in it (φ1, . . . , φn, I1, . . . , In)
+and the standard symplectic form ω in it. And assume the Hamiltonian function
+of the system is given by h(I) a function only depending on the action coordinates.
+Then the Hamilton equations of the system are given by
+ιXhω = dh
+where Xh is the vector field generating the trajectories. Because h does not de-
+pend on φ the angular variables the system is really easy to solve, and the equations
+are given by
+x(t) = (φ(t), I(t)) = (φ0 + ut, I0),
+where u = ∂h/∂I is called the frequency vector. These motions for a fixed
+initial condition are inside a Liouville torus, and are called quasi-periodic.
+The KAM theorem studies what happens to such systems when a small per-
+turbation is applied to the Hamiltonian function, i.e. we consider the evolution of
+the system given by the Hamiltonian h(I) + R(I, φ), where we think of the term
+R(I, φ) as the small perturbation in the system. With this in mind, the Hamilton
+equations can be written as
+˙φ = u(I) + ∂
+∂I R(I, φ), ˙I = − ∂
+∂φR(I, φ),
+Another important concept to have in mind is the concept of rational depen-
+dency. A frequency u is rationally dependent if ⟨u, k⟩ = 0 for some k ∈ Zn, if there
+exists no k satisfying the condition then the vector u is called rationally indepen-
+dent. There is a stronger concept of being rationally independent and that is the
+concept of being Diophantine. A vector u is γ,τ-diophantine if ⟨u, k⟩ ≥
+γ
+|k|τ
+1 for all
+k ∈ Zn \ {0}. γ > 0 and τ > n − 1.
+The KAM theorem states that the Liouville tori with frequency vector satisfying
+the diophantine condition survive under the small perturbation R(I, φ). There are
+conditions relating the size of the perturbation with γ and τ. Also, the set of tori
+satisfying the Diophantine condition has measure 1 − Cγ for some constant C.
+Now we give a proper statement of the theorem as was given in [DG96].
+15
+
+16
+3. A CRASH COURSE ON KAM THEORY
+Theorem 3.1 (Isoenergetic KAM theorem). Let G ⊂ Rn, n > 2, a compact,
+and let H(φ, I) = h(I) + f(φ, I) real analytic on Dρ(G).
+Let ω = ∂h/∂I, and
+assume the bounds:
+����
+∂2h
+∂I2
+����
+G,ρ2
+≤ M,
+|ω|G ≤ L
+and
+|ωn(I)| ≥ l∀I ∈ G.
+Assume also that ω is µ-isoenergetically non-degenerate on G. For a = 16M/l2,
+assume that the map Ω = Ωω,h,a is one-to-one on G, and that its range F = Ω(G)
+is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1 given, and assume:
+ε := ∥f∥G,ρ ≤ ν2l6µ2ˆρ2τ+2
+24τ+32L6M 3 · γ2,
+γ ≤ min
+�8LMρ2
+νlˆρτ+1 , l
+�
+,
+where we write ρ := min
+�
+νρ1
+12(τ+2), 1
+�
+. Define the set
+ˆG = ˆGγ :=
+�
+I ∈ G − 2γ
+µ : ω(I)isτ, γ − Diophantine
+�
+.
+Then, there exists a real continuous map T : W ρ1
+4 (Tn) × ˆG → Dρ(G), analytic
+with respect to the angular variables, such that:
+(1) For every I ∈ ˆG, the set T (Tn × {I}) is an invariant torus of H, its
+frequency vector is colinear to ω(I) and its energy is h(I).
+(2) Writing
+T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)),
+one has the estimates
+|Tφ| ˆ
+G,( ρ1
+4 ,0),∞ ≤ 22τ+15L2M
+ν2l2ˆρ2τ+1
+ε
+γ2 ,
+|TI| ˆ
+G,( ρ1
+4 ,0) ≤ 2τ+16L3M
+νl3µˆρτ+1
+ε
+γ
+(3) meas[(Tn ×G)\T (Tn × ˆG)] ≤ Cγ, where C is a very complicated constant
+depending on n, τ, diamF, D, ˆρ, M, L, l, µ.
+Remark 3.2. This version of the KAM theorem is the isoenergetic one, this
+version ensures that the energy of the Liouville Tori identified by the diffeomorphism
+after the perturbation remains the same as before the perturbation. Our version of
+the bm-KAM is not isoenergetic for the sake of simplifying the computations.
+Also, we should outline that the KAM theorem has already been explored in
+singular symplectic manifolds before. In [KMS16a] the authors proved a KAM
+theorem for b-symplectic manifolds, for a particular kind of perturbations.
+Theorem 3.3 (KAM Theorem for b-Poisson manifolds). Let Tn × Bn
+r be en-
+dowed with standard coordinates (ϕ, y) and the b-symplectic structure. Consider a
+b-function
+H = k log |y1| + h(y)
+on this manifold, where h is analytic. Let y0 be a point in Bn
+r with first component
+equal to zero, so that the corresponding level set Tn × {y0} lies inside the critical
+hypersurface Z.
+Assume that the frequency map
+˜ω : Bn
+r → Rn−1,
+˜ω(y) := ∂h
+∂˜y (y)
+
+3. A CRASH COURSE ON KAM THEORY
+17
+has a Diophantine value ˜ω := ˜ω(y0) at y0 ∈ Bn and that it is non-degenerate at y0
+in the sense that the Jacobian ∂˜ω
+∂˜y (y0) is regular.
+Then the torus Tn × {y0} persists under sufficiently small perturbations of H
+which have the form mentioned above, i.e. they are given by ǫP, where ǫ ∈ R and
+P ∈b C∞(Tn × Bn
+r ) has the form
+P(ϕ, y) = k′ log |y1| + f(ϕ, y)
+f(ϕ, y) = f1( ˜ϕ, y) + y1f2(ϕ, y) + f3(ϕ1, y1).
+More precisely, if |ǫ| is sufficiently small, then the perturbed system
+Hǫ = H + ǫP
+admits an invariant torus T .
+Moreover, there exists a diffeomorphism Tn → T close1 to the identity taking
+the flow γt of the perturbed system on T to the linear flow on Tn with frequency
+vector
+�k + ǫk′
+c
+, ˜ω
+�
+.
+1By saying that the diffeomorphism is “ǫ-close to the identity” we mean that, for given H, P
+and r, there is a constant C such that ∥ψ − Id∥ < Cǫ.
+
+
+Part 2
+Action-angle coordinates and
+cotangent models
+
+In this part, we consider the semilocal classification for any bm-Poisson manifold
+in a neighbourhood of an invariant compact submanifold. The compact subman-
+ifolds under consideration are the compact invariant leaves of the distribution D
+generated by the Hamiltonian vector fields Xfi of an integrable system. An in-
+tegrable system is given by a set of n functions on a 2n-dimensional symplectic
+manifold which we can order in a map F = (f1, . . . , fn). Historically, integrable
+systems were introduced to actually integrate Hamiltonian systems XH using the
+first-integrals fi and, classically, we identify H = f1. It turns out that in the sym-
+plectic context the compact regular orbits of the distribution D coincide with the
+fibers F −1(F(p)) for any point p on these orbits/fibers. The fact that the orbit
+coincides with the connected fiber is part of the magic of symplectic duality.
+The same picture is reproduced for singular symplectic manifolds of bm-type
+or bm-Poisson manifolds as we will see in this chapter.
+The study of action-angle coordinates has interest from this geometrical point
+of view of the classification of geometric structures in a neighbourhood of a compact
+submanifold of a bm-Poisson manifold. It also has interest from a dynamical point
+of view as these compact submanifolds now coincide with invariant subsets of the
+Hamiltonian system under consideration.
+From a geometric point of view, the existence of action-angle coordinates deter-
+mines a unique geometrical model for the bm-Poisson (or bm-symplectic) structure
+in a neighbourhood of the invariant set. From a dynamical point of view, the exis-
+tence of action-angle coordinates provides a normal form theorem that can be used
+to study stability and perturbation problems of the Hamiltonian systems (as we
+will see in the last chapter of this monograph).
+An important ingredient that makes our action-angle coordinate theorem brand-
+new from the symplectic perspective is that the system under consideration is more
+general than Hamiltonian, it is bm-Hamiltonian as the first-integrals of the system
+can be bm-functions which are not necessarily smooth functions. Dynamically, this
+means that we are adding to the set of Hamiltonian invariant vector fields, the
+modular vector field of the integrable system.
+In contrast to the standard action-angle coordinates for symplectic manifolds,
+our action-angle theorem comes with m additional invariants associated with the
+modular vector field which can be interpreted in cohomological terms as the pro-
+jection of the bm-cohomology class determined by the modular vector field on the
+first cohomology group of the critical hypersurface under the Mazzeo-Melrose cor-
+respondence.
+The strategy of the proof of the action-angle coordinate systems is the search
+of a toric action (so this takes us back to the motivation of the use of symmetries
+in this monograph). In contrast to the symplectic case, it is not enough that this
+action is Hamiltonian as then a direction of the Liouville torus would be missing.
+We need the toric action to be bm-Hamiltonian. The structure of this proof looks
+like the one in [KMS16a] but encounters serious technical difficulties as in order
+to check that the natural action to be considered is bm-Hamiltonian we need to go
+deeper inspired by [Sco16] in the relation between the geometry of the modular
+vector field and the coefficients of the Taylor series ci of one of the first-integrals.
+This allows us to understand new connections between the geometry and analysis
+of bm-Poisson structures not explored before.
+
+21
+Once we prove the existence of this bm-Hamiltonian action the proof looks very
+close to the one in [KMS16a].
+In the second chapter of this part we re-state the action-angle theorem as a
+cotangent lift theorem with the following mantra:
+Every integrable system on a bm-Poisson manifold looks like a bm-cotangent lift
+in a neighborhood of a Liouville torus.
+
+
+CHAPTER 4
+An action-angle theorem for bm-symplectic
+manifolds
+4.1. Basic definitions
+4.1.1. On bm-functions. The definition of the analogue of b-functions in the
+bm-setting is somewhat delicate. The set of bmC∞(M) needs to be such that for all
+the functions f ∈bm C∞(M), its differential df is a b-form, where d is the bm-exterior
+differential. Recall that a form in bmΩk(M) can be locally written as
+α ∧ dx
+xm + β
+where α ∈ Ωk−1(M) and β ∈ Ωk(M). Recall also that
+d
+�
+α ∧ dx
+xm + β
+�
+= dα ∧ dx
+xm + dβ.
+We need df to be a well-defined bm-form of degree 1. Let f = g
+1
+xk−1 , then
+df = dg
+1
+xk−1 − g k−1
+xk dx. This from can only be a bm-form if and only if g only
+depends on x. If f = g log(x), then dg log(x) + g 1
+xdx, which imposes dg = 0 and
+hence g to be constant.
+With all this in mind, we make the following definition.
+Definition 4.1. The set of bm-functions is defined recursively according to the
+formula
+bmC∞(M) = x−(m−1)C∞(x) + bm−1C∞(M)
+with C∞(x) the set of smooth functions in the defining function x and
+bC∞(M) = {g log |x| + h, g ∈ R, h ∈ C∞(M)}.
+Remark 4.2. A bmC∞(M)-function can be written as
+f = a0 log x + a1
+1
+x + . . . + am−1
+1
+xm−1 + h
+where ai, h ∈ C∞(M).
+Remark 4.3. From this chapter on we are only considering bm-manifolds
+(M, x, Z) with x defined up to order m.
+I.e.
+we can think of x as a jet of a
+function that coincides up to order m to some defining function. This is the orig-
+inal viewpoint of Scott in [Sco16] which we adopt from now on. The difference
+with respect to the other chapters is that we do not fix an specific function.
+Definition 4.4. We say that two bm-integrable systems F1, F2 are equivalent
+if there exists ϕ, a bm-symplectomorphism, i.e. a diffeomorphism preserving both
+ω and the critical set Z (“up to order m”1), such that ϕ ◦ F1 = F2.
+1I.e. it preserves the jet x
+23
+
+24
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+Remark 4.5. The Hamiltonian vector field associated to a bm-function f is a
+smooth vector field. Let us compute it locally using the bm-Darboux theorem:
+Π = xm
+1
+∂
+∂x1
+∧
+∂
+∂y1
++
+m
+�
+i=2
+∂
+∂xi
+∧ ∂
+∂yi
+and f = a0 log x1 +
+m−1
+�
+i=1
+ai
+1
+xi
+1
++ h.
+Then if we compute
+df
+=
+c1
+����
+a0
+1
+x1
++
+m−1
+�
+i=1
+ci
+�
+��
+�
+(a′
+i − (i − 1)ai−1) 1
+xi
+1
+dx1
+−
+cm
+�
+��
+�
+(m − 1)am−1
+1
+xm
+1 dx1 + dh
+=
+m
+�
+i=1
+ci
+xi
+1
+dx1 + dh.
+Then,
+(4.1)
+Xf = Π(df, ·) =
+m
+�
+i=1
+cixm−i
+1
+∂
+∂y1
++ Π(dh, ·),
+we obtain a smooth vector field.
+4.2. On bm-integrable systems
+In this section we present the definition of a bm-integrable system as well as
+some observations about these objects.
+Definition 4.6. Let (M 2n, Z, x) be a bm-manifold, and let Π be a bm-Poisson
+structure on it. F = (f1, . . . , fn)2 is a bm-integrable system3 if:
+(1) df1, . . . , dfn are independent on a dense subset of M and in all the points
+of Z where independent means that the form df1 ∧ . . . ∧ dfn is non-zero as
+a section of Λn(bmT ∗(M)),
+(2) the functions f1, . . . , fn Poisson commute pairwise.
+Definition 4.7. The points of M where df1, . . . , dfn are independent are called
+regular points.
+The next remarks will lead us to a normal form for the first function f1.
+Remark 4.8. Note that df1, . . . , dfn are independent on a point if and only if
+Xf1, . . . , Xfn are independent at that point. This is because the map
+bmT M →bm T ∗M : u �→ ωp(u, ·)
+is an isomorphism.
+Remark 4.9. The condition of df1, . . . , dfn being independent must be under-
+stood as df1 ∧ . . . ∧ dfn being a non-zero section of �n( bmT ∗M).
+2fi are bm-functions.
+3In this monograph we only consider integrable systems of maximal rank n.
+
+4.2. ON bm-INTEGRABLE SYSTEMS
+25
+Remark 4.10. By remark 4.8 the vector fields Xf1, . . . , Xfn have to be in-
+dependent.
+This implies that one of the f1, . . . , fn has to be a singular (non-
+smooth) bm-function with a singularity of maximal degree.
+If we write fi =
+c0,i log(x1) + �m−1
+j=1
+cj,i
+xj
+1 + ˜f1
+Xfi =
+m
+�
+j=1
+xm−j
+1
+ˆcj,i
+∂
+∂y1
++ X ˜
+fi
+where ˆcj,i(x) = d(cj,i)
+dx
+− (j − 1)cj−1,i. If there is no bm-function with a singularity
+of maximum degree all the terms in the ∂/∂y1 direction become 0 at Z. And hence
+Xf1, . . . , Xfn cannot have maximal rank at Z.
+Lemma 4.11. Let F = (f1, . . . , fn) a bm-integrable system. If f1 has a singular-
+ity of maximal degree, there exists an equivalent integrable system F ′ = (f ′
+1, . . . , f ′
+n)
+where f ′
+1 has a singularity of maximal degree and no other f ′
+i has singularity of
+any degree.
+Proof. Let fi = c0,i log(x1) +
+m−1
+�
+j=1
+cj,1
+xj
+1
+�
+��
+�
+ζi(x1)
++ ˜fi = ζi(x1) + ˜fi. By remark 4.104,
+Xfi =
+m
+�
+i=1
+xm−j
+1
+ˆcj,i
+�
+��
+�
+gi(x1)
+∂
+∂y1
++ X ˜
+fi = gi(x1) ∂
+∂y1
++ X ˜
+fi.
+Note that gi(x1) = gi(0) = ˆcm,i at Z. Let us look at the distribution given by the
+Hamiltonian vector fields Xfi = gi(x1) ∂
+∂y1 +X ˜
+fi. This distribution is the same that
+the one given by:
+(4.2)
+{Xf1, Xf2 − g2(x1)
+g1(x1)Xf1, . . . , Xfn − gn(x1)
+g1(x1) Xf1}.
+Observe that for i > 1, Xfi − gi(x1)
+g1(x1)Xf1 = X ˜
+fi + g2(x1)
+g1(x1)X ˜
+f1. Also g1(x1) is different
+from 0 close to Z because at Z g1(x1) = ˆcm,1. Since the distribution given by these
+vector fields is the same, an integrable system that has Hamiltonian vector fields 4.2
+would be equivalent to F. From the expression 4.2 it is clear that the new vector
+fields commute. And it is also true that this new vector fields are Hamiltonian. Let
+us take F ′ the set of functions that have as Hamiltonian vector fields 4.2.
+□
+From now on we will assume the integrable system to have only one singular
+function and this function to be f1.
+Remark 4.12. Because we asked Xf1, . . . , Xfn to be linearly independent at
+all the points of Z and using the previous remarks cm := cm,1 ̸= 0 at all the points
+of Z.
+4Here have used the bm-Darboux theorem to do the computations.
+
+26
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+Furthermore, we can assume f1 to have a smooth part equal to zero as sub-
+tracting the smooth part of f1 to all the functions gives an equivalent system. Also,
+we can assume that cm is 1 because dividing all the functions of the bm-integrable
+system by cm also gives us an equivalent system.
+As a summary, we can assume f1 = a0 log(x)+a11/x+. . .+am−21/xm−2+
+1/xm−1 and f2, . . . , fn to be smooth, a0 ∈ R and a1, . . . , am−2 ∈ C∞(x).
+Also we are going to state lemma 3.2 in [GMPS17], because we are going to
+use it later in this section. The result states that if we have a toric action on a bm-
+symplectic manifold (which we will prove in a neighbourhood of a Liouville torus),
+then we can assume the coefficients a2, . . . , am−2 to be constants. More precisely
+Lemma 4.13. There exists a neighborhood of the critical set U = Z × (−ε, ε)
+where the moment map µ : M → t∗ is given by
+µ = a1 log |x| +
+m
+�
+i=2
+ai
+x−(i−1)
+i − 1
++ µ0
+with ai ∈ t0
+L and µ0 is the moment map for the TL-action on the symplectic leaves
+of the foliation.
+4.3. Examples of bm-integrable systems
+The following example illustrates why it is necessary to use the definition of bm-
+function as considered above. There are natural examples of changes of coordinates
+in standard integrable systems on symplectic manifolds that yield bm-symplectic
+manifolds but do not give well-defined bm-integrable systems.
+Example 4.14. Consider a time change in the two body problem, to obtain
+a b2-integrable system. In the classical approach to solve the 2-body problem the
+following two conserved quantities are obtained:
+f1
+=
+µy2
+2 +
+l2
+2µr2 − k
+r ,
+f2
+=
+l,
+with symplectic form ω = dr ∧ dy + dl ∧ dα, where r is the distance between the
+two masses and l is the angular momentum. We also know that l is constant along
+the trajectories. Because l is a constant of the movement, we can do a symplectic
+reduction on its level sets. The form on the symplectic reduction becomes dr ∧ dy.
+To simplify the notation, we will use x instead of r. Then ω = dx ∧ dy. With
+hamiltonian function given by f = µ
+2 y2 +
+l
+2µ
+1
+x2 − k 1
+x. Hence, the equations are:
+˙x
+=
+∂f
+∂y ,
+˙y
+=
+− ∂f
+∂x.
+Doing a time change τ = x3t then dx
+dτ =
+1
+x3 dx
+dt . With this time coordinate, the
+equations become:
+˙x
+=
+1
+x3
+∂f
+∂y ,
+˙y
+=
+− 1
+x3
+∂f
+∂x.
+These equations can be viewed as the motion equations given by a b3-symplectic
+form ω =
+1
+x3 dx ∧ dy.
+Let us check that this is actually a bm-integrable system.
+
+4.3. EXAMPLES OF bm-INTEGRABLE SYSTEMS
+27
+• All the functions Poisson commute is immediate because we only have
+one.
+• df = µydy +( k
+x2 − l
+µ
+1
+x3 )dx is a b3-form because the term with dx does not
+depend on y.
+• All the functions are independent, this is true because df does not vanish
+as a b3-form.
+Example 4.15. In the paper [Mar19] the author builds an action of SL(2, R)
+over (P, ωP ) where P = {ξ ∈ C|i(¯ξ − ξ) > 0} is the complex semi-plane, with
+moment map JP (ξ) =
+R
+ξim ((|ξ|2 + 1), 2ξr, ±(|ξ|2 + 1)), where the ± sign depends
+on the choice of the hemisphere projected by the stereographic projection. From
+now on we will take the sign +. Also the symplectic form ωP has the following
+expression:
+ωP = ± R
+ξ2
+im
+dξr ∧ dξim
+In order to simplify the notation we identify P with the real half-plane P =
+{x, y ∈ R2|y > 0}. With this identification, the moment map becomes Jp(x, y) =
+R
+y (x2 + y2 + 1, 2x, x2 + y2 + 1). Obviously, this moment map does not give an
+integrable system. The symplectic form writes as:
+ωP = R
+y2 dy ∧ dx.
+This form can be viewed as a b2-form if we extend P including the line {y = 0}
+as its singular set.
+Let us consider only one of the components of JP as bm-
+function and let us see if it gives a bm-integrable system. First we will try with
+f1 = R
+y (x2 + y2 + 1) and then f2 = R
+y (2x).
+(1) f1 = R
+y (x2 + y2 + 1) We have to check three things to see if this gives a
+b2-integrable system.
+(a) All the functions Poisson commute is immediate because we only
+have one.
+(b) All the functions are bm-functions. This point does not hold because
+df1 = R
+y2 (2xydx + (y2 − x2 − 1)dy) and the first component makes no
+sense as a section of Λ1(b2T ∗M).
+(c) All the functions are independent. In this case, we need to check that
+df1 does not vanish, but since it is not a bm-form it makes no sense
+to be a non-zero section of Λ1(b2T ∗M).
+(2) f2 = R
+y (2x)
+(a) Same as before.
+(b) All the functions are bm-functions. This point does not hold because
+df2 = 2R
+y dx − 2Rx
+y2 dy and the first component makes no sense as a
+section of Λ1(b2T ∗M).
+(c) Same as before.
+Example 4.16. Toric actions give natural examples of integrable systems where
+the component functions are given by the moment map. In the case of surfaces:
+S1-actions on surfaces give natural examples of bm-integrable systems. Only torus
+and spheres admit circle actions.
+
+28
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+In the picture below two integrable systems on the 2-sphere depending on the
+degree m. On the right the image of the moment map that defines the integrable
+system. The action is by rotations along the central axis.
+Namely consider the sphere S2 as a bm-symplectic manifold having as critical
+set the equator:
+(S2, Z = {h = 0}, ω = dh
+hm ∧ dθ),
+with h ∈ [−1, 1] and θ ∈ [0, 2π).
+For m = 1: The computation ι ∂
+∂θ ω = − dh
+h = −d(log |h|), tells us that the
+function µ(h, θ) = log |h| is the moment map and defines a b-integrable system.
+For higher values of m: ι ∂
+∂θ ω = − dh
+hm = −d(−
+1
+(m−1)hm−1 ), and the moment
+map is µ(h, θ) = −
+1
+(m−1)hm−1 which defines a bm-integrable system.
+µ, m = 1
+µ, m = 2
+Figure 1. Integrable systems associated to the moment map of
+an S1-action by rotations on a bm-symplectic 2-sphere S2.
+Example 4.17. Consider now as b2-symplectic manifold the 2-torus
+(T2, Z = {θ1 ∈ {0, π}}, ω =
+dθ1
+sin2 θ1
+∧ dθ2)
+with standard coordinates: θ1, θ2 ∈ [0, 2π). Observe that the critical hypersurface
+Z in this example is not connected. It is the union of two disjoint circles. Consider
+the circle action of rotation on the θ2-coordinate with fundamental vector field
+∂
+∂θ2 .
+As the following computation holds,
+ι
+∂
+∂θ2 ω = − dθ1
+sin2 θ1
+= d
+�cos θ1
+sin θ1
+�
+.
+The fundamental vector field of the S1-action defines b2C∞-integrable system given
+by the function − cos θ1
+sin θ1 .
+Example 4.18. The former example can be made general to produce examples
+of bm-integrable systems on a bm-symplectic manifold for any integer m
+(T2, Z = {θ1 ∈ {0, π}}, ω =
+dθ1
+sinm θ1
+∧ dθ2).
+Then
+ι
+∂
+∂θ2 ω = −
+dθ1
+sinm θ1
+= d
+�
+| cos θ1|
+cos θ1
+2F1
+� 1
+2, 1−m
+2 ; 3−m
+2 ; sin2(θ1)
+�
+(1 − m) sinm−1 θ1
+�
+,
+with 2F1 the hypergeometric function.
+
+4.4. LOOKING FOR A TORIC ACTION
+29
+µ
+Figure 2. Integrable system given by an S1-action on a b2-torus
+T2 and its associated moment map.
+Thus, the associated S1-action has as bmC∞-Hamiltonian the function
+−| cos θ1|
+cos θ1
+2F1
+� 1
+2, 1−m
+2 ; 3−m
+2 ; sin2(θ1)
+�
+(1 − m) sinm−1 θ1
+which defines a bm-integrable system.
+Now we give a couple of examples of bm-integrable systems.
+Example 4.19. This example uses the product of bm-integrable systems on a
+bm-symplectic manifold with an integrable system on a symplectic manifold. Given
+(M 2n1
+1
+, Z, x, ω1) a bm-symplectic manifold with f1, . . . , fn1 a bm-integrable system
+and (M 2n2
+2
+, ω2) a symplectic manifold with g1, . . . , gn2 an integrable system. Then
+(M1×M2, Z×M2, x, ω1+ω2) is a bm-symplectic manifold and (f1, . . . , fn1, g1, . . . , gn2)
+is a bm-integrable system on the higher dimensional manifold.
+In particular by combining the former examples of bm-integrable systems on
+surfaces and arbitrary integrable systems on symplectic manifolds we obtain exam-
+ples of bm-integrable systems in any dimension.
+Example 4.20. (From integrable systems on cosymplectic manifolds
+to bm-integrable systems:)
+Using the extension theorem (Theorem 50) of [GMP14] we can extend any
+integrable system (f2, . . . , fn) to an integrable system in a neighbourhood of a
+cosymplectic manifold (Z, α, ω) by just adding a bm-function f1 to the integrable
+system so that the new integrable system is (f1, f2, . . . , fn) and considering the
+associated bm-symplectic form:
+(4.3)
+˜ω = p∗α ∧ dt
+tm + p∗ω.
+(t is the defining function of Z).
+4.4. Looking for a toric action
+In this section we pursue the proof of action-angle coordinates for bm-integrable
+systems by recovering a torus group action. This action is associated to the Hamil-
+tonian vector fields associated to Xfi.
+This is the same strategy used for b-integrable systems in [KMS16a]- One of
+the main difficulties is to prove that the coefficients a1, . . . , an can be considered
+
+30
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+as constant functions. This makes it more difficult to prove the existence of a Tn-
+action in the general bm-case than in the b-case, but once we have it we can use
+the results in [GMW17] to assume that the coefficients a1, . . . , an are constant
+functions.
+In this section we provide some preliminary material that will be needed later:
+Proposition 4.21. Let (M, Z, x, ω) be a bm-symplectic manifold such that Z
+is connected with modular period k. Let π : Z → S1 ≃ R/kZ be the projection
+to the base of the corresponding mapping torus. Let γ : S1 = R/kZ → Z be any
+loop such that π ◦ γ is positively oriented and has constant velocity 1. Then the
+following are equal:
+(1) The modular period of Z,
+(2)
+�
+γ ιLω,
+(3) The value am−1 for any bmC∞(M)-function
+f = a0 log(x) +
+m−1
+�
+j=1
+aj
+1
+xj + h
+such that the hamiltonian vector field Xf has 1-periodic orbits homotopic
+in Z to some γ.
+Proof. Let us first prove that (1)=(2) and then that (2)=(3).
+(1)=(2) Let us denote by Vmod the modular vector field. Recall from [GMW17]
+that ιL(Vmod) is the constant function 1. Let s : [0, k] → Z be the trajec-
+tory of the modular vector field. Because the modular period is k, s(0)
+and s(k) are in the same leaf L. Let ˆs : [0, k + 1] → Z a smooth extension
+of s such that s|[k,k+1] is a path in L joining ˆs(k) = s(k) to ˆs(k+1) = s(0).
+This way ˆs becomes a loop. Then,
+k =
+� k
+0
+1dt =
+�
+S
+ιLω =
+�
+ˆs
+ιLω =
+�
+γ
+ιLω
+(2)=(3) Let r : [0, 1] �→ Z be the trajectory of Xf the hamiltonian vector field of
+f. Recall that Xf satisfies
+ιXf ω =
+m
+�
+j=1
+cj
+dx
+xi + dh.
+Let xm ∂
+∂x be a generator of the linear normal bundle L. We know that
+Xf is 1-periodic and its trajectory is homotopic to γ. Hence,
+k =
+�
+r ιLω
+=
+� 1
+0
+ιxm ∂
+∂x ω(Xf|r(t))dt
+=
+� 1
+0
+−(
+m
+�
+j=1
+ci
+dx
+xi + dh) · (xm ∂
+∂x)|r(t)dt
+=
+−cm = −am−1
+□
+
+4.4. LOOKING FOR A TORIC ACTION
+31
+We will also need a Darboux-Carath´eodory theorem for bm-symplectic mani-
+folds:
+Theorem 4.22 (Darboux-Carath´eodory (bm-version)). Let
+(M 2n, x, Z, ω)
+be a bm-symplectic manifold and m be a point on Z. Let f1, . . . , fn be a bm-integrable
+system. Then there exist bm-functions (q1, . . . , qn) around m such that
+ω =
+n
+�
+i=1
+dfi ∧ dqi
+and the vector fields {Xfi, Xqj}i,j commute. If f1 is not smooth (recall that f1 =
+a0 log(x) + �m−1
+j=1 aj 1
+xi with an ̸= 0 on Z and a0 ∈ R) the qi can be chosen to be
+smooth functions, and (x, f2, . . . , fn, q1, . . . , qn) is a system of local coordinates.
+Proof. The first part of this proof is exactly as in [KMS16a]. Assume now
+f1 = a0 log(x) +
+m−1
+�
+j=1
+aj
+1
+xi . We modify the induction requiring also that µi (in
+addition to be in Ki) is also in T ∗M ⊆b T ∗M. We can also ask this extra condition
+while asking µi(Xfi) = 1, we only have to check that Xfi does not vanish in T M.
+This is clear because Xfi does not vanish at bT M and
+0 = {fn, fi} =
+� m
+�
+i=1
+˜ai
+dx
+xi
+�
+(Xfi) =
+�
+dx
+xm
+m
+�
+i=1
+aixi
+�
+(Xfi).
+All the terms in the last expression vanish except for the one of degree m.
+Then dx/xm is in the kernel of Xfi, hence Xfi does not vanish on T M and the
+qi can be chosen to be smooth.
+{Xx, Xf2, . . . , Xfn, Xq1, . . . Xqn} commute because {Xfi, Xqi}i,j commute. Then
+dx ∧ df2 . . . ∧ dfn ∧ dq1 ∧ . . . ∧ dqn
+is a non-zero section of �n(bT M). And hence
+(x, f2, . . . , fn−1, q1, . . . , qn)
+are local coordinates.
+□
+Before proceeding with the proof of the action-angle coordinates, we need to
+prove that in a neighbourhood of a Liouville torus the fibration is semilocally trivial:
+Lemma 4.23 (Topological Lemma). Let m ∈ Z be a regular point of a bm-
+integrable system (M, x, Z, ω, F). Assume that the integral manifold Fm through m
+is compact. Then there exists a neighborhood U of Fm and a diffeomorphism
+φ : U ≃ Tn × Bn
+which takes the foliation F to the trivial foliation {Tn × {b}}b∈Bn.
+Proof. We follow the steps of [LGMV08]. In this case, the only extra step
+that must be checked is that the foliation given by the bm-hamiltonian vector fields
+of F = (f1, f2, . . . , fn) is the same as the one given by the level sets of ˜F :=
+(x, f2, . . . , fn). In our case f1 = a0 log(x) + �m−1
+u=1 ai 1
+xi , where a0 ∈ R, ai ∈ C∞(x),
+am−1 = 1. Hence the foliations are the same. Then as in [LGMV08], we take an
+
+32
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+Figure 4. Fibration by Liouville tori: The middle fiber of the
+point p ∈ Z in magenta, the neighbouring Liouville tori in blue.
+arbitrary Riemannian metric on M and this defines a canonical projection ψ : U →
+Fm. Let us define φ := ψ × ˜F. We obtain the commutative diagram (Figure 3).
+U
+Tn × Bn
+Bn
+φ
+˜
+F
+p
+Figure 3. Commutative diagram of the construction of the iso-
+morphism of bm-integrable systems.
+which provides the necessary equivalence of bm-integrable systems.
+□
+4.5. Action-angle coordinates on bm-symplectic
+manifolds
+In a neighbourhood of one of our Liouville tori all we can assume about the
+form of our bm-symplectic structure is that is given by the Laurent series defined
+in [Sco16].
+That is to say, we can assume that in a tubular neighborhood U of Z
+ω =
+m−1
+�
+j=1
+dx
+xi ∧ π∗(αi) + β,
+where π : U → Z is the projection of the tubular neighborhood onto Z, αi are
+closed smooth de Rham forms on Z and β a de Rham form on M of degree 2.
+In [RBM, MM22] normal forms are given for group actions in a neighbour-
+hood of the orbit. Below we provide a normal for the integrable system in a neigh-
+bourhood of an orbit of the torus action associated to the integrable system. This
+theorem is finer than the bm-symplectic slice theorem provided in [MM22] as it
+also gives information about the first integrals.
+One of the non-trivial steps of the proof is to associate a toric action to the
+integrable system. The connection to normal forms of group actions will become
+even more evident when we discuss the associated cotangent models.
+
+4.5. ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC
+MANIFOLDS
+33
+Theorem A (Action-angle coordinates for bm-symplectic manifolds). Let (M, x, ω, F)
+be a bm-integrable system, where F = (f1 = a0 log(x) + �m−1
+j=1 aj 1
+xj , . . . , fn) with
+aj for j > 1 functions in x. Let m ∈ Z be a regular point and let us assume that the
+integral manifold of the distribution generated by the Xfi through m is compact.
+Let Fm be the Liouville torus through m. Then, there exists a neighborhood U of
+Fm and coordinates (θ1, . . . , θn, σ1, . . . , σn) : U → Tn × Bn such that:
+(1) We can find an equivalent integrable system F = (f1 = a′
+0 log(x) +
+�m−1
+j=1 a′
+j
+1
+xj , . . . , fn) such that the coefficients a′
+0, . . . , a′
+m−1 of f1 are con-
+stants ∈ R,
+(2)
+ω|U =
+
+
+m
+�
+j=1
+c′
+j
+c
+σj
+1
+dσ1 ∧ dθ1
+
+ +
+n
+�
+i=2
+dσi ∧ dθi
+where c is the modular period and c′
+j = −(j − 1)a′
+j−1, also
+(3) the coordinates σ1, . . . , σn depend only on f1, . . . fn.
+Proof. The idea of this proof is to construct an equivalent bm-integrable
+system whose fundamental vector fields define a Tn-action on a neighborhood of
+Tn × {0}. It is clear that all the vector fields Xf1, . . . , Xfn define a torus action on
+each Liouville tori Tn × {b} where b ∈ Bn, but this does not guarantee that their
+flow defines a toric action on all Tn × Bn. The proof is structured in three steps.
+The first one is the uniformization of the periods, i.e. we define an Rn-action on a
+neighborhood of Tn × {0} such that the lattice defined by its kernel at every point
+is constant. This allows to induce an actual action of a torus (as the periods are
+constant) of rank n: A Tn action by taking quotients. The second step consists
+in checking that this action is actually bm-Hamiltonian. And in the final step we
+apply theorem 4.22 to obtain the expression of ω.
+(1) Uniformization of periods.
+Let Φs
+XF be defined as the joint flow by the Hamiltonian vector fields
+of the action:
+(4.4)
+Φ : Rn × (Tn × Bn)
+→
+(Tn × Bn)
+((s1, . . . , sn), (x, b))
+�→
+Φs1
+Xf1 ◦ · · · ◦ Φsn
+Xfn ((x, b))
+this defines an Rn-action on Tn×Bn. For each b ∈ Bn at a single orbit
+Tn × {b} the kernel of this action is a discrete subgroup of Rn. We will
+denote the lattice given by this kernel Λb. Because the orbit is compact,
+the rank of Λb is maximal i.e. n. This lattice is known as the period lattice
+of Tn × {b} as we know by standard arguments in group theory that the
+lattice has to be of maximal rank so as to have a torus as a quotient. In
+general we can not assume that Λb does not depend on b. The process of
+uniformization of the periods modifies the action 4.4 in such a way that
+Λb = Zn for all b. Let us consider the following Hamiltonian vector field
+�n
+i=1 kiXfi. The bm-function that generates this Hamiltonian vector field
+is:
+k1
+
+a0 log(x) +
+m−1
+�
+j=1
+aj
+1
+xj
+
+ +
+n
+�
+i=2
+kifi
+
+34
+4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS
+where recall that am−1 is constant equal 1. Observe that the coefficient
+multiplying 1/xm−1 is k1. By proposition 4.21 k1 = c the modular period.
+In this case c = [αm].
+Hence, for b ∈ Bn−1 × {0} the lattice Λb is contained in Rn−1 × cZ ⊆
+Rn. Pick (λ1, . . . , λn) : Bn → Rn such that:
+• (λ1(b), . . . , λn(b)) is a basis of Λb for all b ∈ Bn,
+• λn
+i vanishes along Bn−1 × {0} at order m for i < n and λi is equal
+to c along Bn−1 × {0}.
+In the previous points, λj
+i denotes the j-th component of λi. The first
+condition can be satisfied by using the implicit function theorem. That
+is because Φ(λ, m) = m is regular with respect to the s coordinates. The
+second condition is automatically true because Λb ⊆ Rn−1×cZ. We define
+the uniformed flow as:
+(4.5)
+˜Φ : Rn × (Tn × Bn)
+→
+(Tn × Bn)
+((s1, . . . , sn), (x, b))
+�→
+Φ(�n
+i=1 siλi, (x, b))
+(2) The Tn-action is bm-Hamiltonian. The objective of this step is to find
+bm-functions σ1, . . . , σn such that Xσi are the fundamental vector fields
+of the Tn-action Yi = �n
+j=1 λj
+iXfj.
+By using the Cartan formula for a bm-symplectic form, we obtain:
+LYiLYiω
+=
+LYi(d(ιYiω) + ιYidω)
+=
+LYi(d(− �n
+j=1 λj
+idfi))
+=
+−LYi(�n
+j=1 dλj
+i ∧ dfj) = 0
+Note that λj
+i are constant on the level sets of F as Φ(λ, m) = m and
+the level sets of F are invariant by Φ.
+Recall that if Y is a complete periodic vector field and P is a bivector
+such that LY LY P = 0, then LY P = 0. So, the vector fields Yi are Poisson
+vector fields. To show that each ιYiω has a bmC∞ primitive we will see
+that [ιYiω] = 0 in the bm-cohomology.
+One one hand, if i > 1, ιYiω vanishes at Z. This holds because Yi has
+not any component ∂/∂Y .
+Recall Proposition 6 from [GMP14]:
+Proposition 4.24. If ω ∈b Ω(M) with ω|Z = 0, then ω ∈ Ω(M).
+In a similar way for bm-forms we have,
+Proposition 4.25. If ω ∈bm Ω(M) with ω|Z vanishing up to order
+m, then ω ∈ Ω(M).
+Thus as ιYiω vanishes at Z, the bm-forms ιYiω are indeed smooth.
+Thus we can now apply the standard Poincar´e lemma and as these forms
+are closed they are locally exact. This proves that all the vector fields Yi
+with i > 1 are indeed Hamiltonian.
+On the other hand, the fact that ιY1ω = cdf1 is obvious.
+Then, because we have a toric action that is Hamiltonian, we can use
+lemma 3.2 in [GMPS17], and we get an equivalent system such that ai
+are all constant and moreover ⟨a′
+i, X⟩ = αi(Xω). Note that by dividing by
+a′
+m−1, we can still assume a′
+m−1 = 1 to be consistent with our notation,
+but we then have to multiply f1 · c in the next step.
+
+4.5. ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC
+MANIFOLDS
+35
+(3) Apply Darboux-Carath´eodory theorem.
+The construction above gives us some candidates σ1 = cf1, σ2, . . . , σn
+for the action coordinates.
+We now apply the Darboux-Carath´eodory theorem and express the
+form in terms of x:
+ω =
+
+
+m
+�
+j=1
+c cj
+xj dx ∧ dq1
+
+ +
+n
+�
+i=2
+dσi ∧ dqi.
+Since the vector fields Xσi =
+∂
+∂qi are fundamental fields of the Tn-
+action the flow 4.5 gives a linear action on the qi coordinates.
+Observe that the coordinate system is only defined in U. It may not
+be valid at points outside U that may be in the orbit of points in U. Let
+us see that the charts can be extended to these points.
+Define U′ the union of all tori that intersect U. We will see that the
+coordinates are valid at U′.
+Let {pi, θj} be the extension of {σi, qj}. It is clear that {pi, θj} = δij
+by its construction in the Darboux-Carath´eodory theorem.
+To see that {θi, θj} = 0 we take the flows by Xpk and extend the
+expression to the whole U′:
+Xpk({θi, θj}) = {{θi, θj}, pk} = {θi, δij} − {θj, δjk} = 0.
+The fact that ω is preserved is obvious because Xpk are hamiltonian
+vector fields and thus they preserve the bm-symplectic forms. Moreover,
+t, θ1, p2, θ2, . . . , pn, θn are independent on U′ and hence are a coordinate
+system in a neighbourhood of the torus.
+□
+Remark 4.26. In the proof we have seen that there exists an equivalent inte-
+grable system where the coefficients of the singular function are indeed constant.
+From now on, when considering a bm-integrable system we are going to make this
+assumption.
+Remark 4.27. By means of the desingularization transformation we may ob-
+tain an action-angle coordinate theorem for folded manifolds as we do in Part 3
+for the KAM theorem for folded symplectic manifolds. This folded action-angle
+theorem is a particular case of the one obtained in [CM22].
+
+
+CHAPTER 5
+Reformulating the action-angle coordinate via
+cotangent lifts
+The action-angle theorem for symplectic manifolds (also known as action-angle
+coordinate theorem) can be reformulated in terms of a cotangent lift.
+Recall that given a Lie group action on any manifold its cotangent lifted action
+is automatically Hamiltonian. By considering the action of a torus on itself by
+translations this action can be lifted to its cotangent bundle and give a semilocal
+normal form theorem as the Arnold-Liouville-Mineur theorem for symplectic man-
+ifolds. If we now replace this cotangent lift to the cotangent bundle to a lift to the
+bm-cotangent bundle we obtain the semilocal normal form of the main theorem of
+this chapter.
+Let start recalling the symplectic and b-symplectic case following [KM17].
+5.1. Cotangent lifts and Arnold-Liouville-Mineur in Symplectic
+Geometry
+Let G be a Lie group and let M be any smooth manifold. Given a group action
+ρ : G × M −→ M, we define its cotangent lift as the action on T ∗M given by
+ˆρg := ρ∗
+g−1 where g ∈ G. We then have a commuting diagram
+T ∗M
+T ∗M
+M
+M
+ˆ
+ρg
+ˆπ
+π
+ρg
+Figure 1. Commutiative diagram of the construction of the iso-
+morphism of bm-integrable systems.
+where π is the canonical projection from T ∗M to M.
+The cotangent bundle T ∗M is a symplectic manifold endowed with the exact
+symplectic form given by the differential of the Liouville one-form ω = −dλ. The
+Lioville one-form can be defined intrinsically:
+(5.1)
+⟨λp, v⟩ := ⟨p, (πp)∗(v)⟩
+with v ∈ T (T ∗M), p ∈ T ∗M.
+A standard argument (see for instance [GS90]) shows that the cotangent lift ˆρ
+is Hamiltonian with moment map µ : T ∗M → g∗ given by
+⟨µ(p), X⟩ := ⟨λp, X#|p⟩ = ⟨p, X#|π(p)⟩,
+37
+
+38
+5. ACTION-ANGLE COORDINATES AND COTANGENT LIFTS
+where p ∈ T ∗M, X is an element of the Lie algebra g and we use the same symbol
+X# to denote the fundamental vector field of X generated by the action on T ∗M
+or M. This construction is known as the cotangent lift.
+In the special case where the manifold M is a torus Tn and the group is Tn
+acting by translations, we obtain the following explicit structure: Let θ1, . . . , θn be
+the standard (S1-valued) coordinates on Tn and let
+(5.2)
+θ1, . . . , θn
+�
+��
+�
+=:θ
+, t1, . . . , tn
+�
+��
+�
+=:t
+be the corresponding chart on T ∗Tn, i.e.
+we associate to the coordinates (5.2)
+the cotangent vector �
+i tidθi ∈ T ∗
+θ Tn. The Liouville one-form is given in these
+coordinates by
+λ =
+n
+�
+i=1
+tidθi
+and its negative differential is the standard symplectic form on T ∗Tn:
+(5.3)
+ωcan =
+n
+�
+i=1
+dθi ∧ dti.
+Denoting by τβ the translation by β ∈ Tn on Tn, its lift to T ∗Tn is given by
+ˆτβ : (θ, t) �→ (θ + β, t).
+The moment map µcan : T ∗Tn → t∗ of the lifted action with respect to the canonical
+symplectic form is
+(5.4)
+µcan(θ, t) =
+�
+i
+tidθi,
+where the θi on the right hand side are understood as elements of t∗ in the obvious
+way. Even simpler, if we identify t∗ with Rn by choosing the standard basis
+∂
+∂θi
+of t then the moment map is just the projection onto the second component of
+T ∗Tn ∼= Tn × Rn. Note that the components of µ naturally define an integrable
+system on T ∗Tn.
+We can rephrase the Arnold-Liouville-Mineur theorem in terms of the symplec-
+tic cotangent model:
+Theorem 5.1. Let F = (f1, . . . , fn) be an integrable system on the symplectic
+manifold (M, ω). Then semilocally around a regular Liouville torus the system is
+equivalent to the cotangent model (T ∗Tn)can restricted to a neighbourhood of the
+zero section (T ∗Tn)0 of T ∗Tn.
+5.2. The case of bm-symplectic manifolds
+Let us start by introducing the twisted bm-cotangent model for torus actions.
+This model has additional invariants: the modular vector field of the connected
+component of the critical set and the modular weights of the associated toric action.
+Consider T ∗Tn be endowed with the standard coordinates (θ, t), θ ∈ Tn, t ∈ Rn and
+consider again the action on T ∗Tn induced by lifting translations of the torus Tn.
+We will now view this action as a bm-Hamiltonian action with respect to a suitable
+bm-symplectic form. In analogy to the classical Liouville one-form we define the
+following non-smooth one-form away from the hypersurface Z = {t1 = 0} :
+
+5.2. THE CASE OF bm-SYMPLECTIC MANIFOLDS
+39
+�
+cc1 log |t1| +
+m
+�
+i=2
+cci
+t−(i−1)
+1
+−(i − 1)
+�
+dθ1 +
+n
+�
+i=2
+tidθi.
+When differentiating this form we obtain a bm-symplectic form on T ∗Tn which
+we call (after a sign change) the twisted bm-symplectic form on T ∗Tn with
+invariants (cc1, . . . , ccm):
+(5.5)
+ωtw,c :=
+
+
+m
+�
+j=1
+cj
+c
+tj
+1
+dt1 ∧ dθ1
+
+ +
+n
+�
+i=2
+dti ∧ dθi,
+where c is the modular period. The moment map of the lifted action is then given
+by
+(5.6)
+µtw,q0,...,qm−1) := (q0 log |t1| +
+m
+�
+i=2
+qit−(i−1)
+1
+, t2, . . . , tn),
+where we are identifying t∗ with Rn and cj = −(j − 1)qj−1.
+We call this lift together with the bm-symplectic form 5.5 the twisted bm-
+cotangent lift with modular period c and invariants (c1, . . . , cm). Note that the
+components of the moment map define a bm-integrable system on (T ∗Tn, ωtw,(cc1,...,ccm)).
+The model of twisted bm-cotangent lift allows us to express the action-angle
+coordinate theorem for bm-integrable systems in the following way:
+Theorem 5.2. Let F = (f1, . . . , fn) be a bm-integrable system on the bm-
+symplectic manifold (M, ω).
+Then semilocally around a regular Liouville torus
+T, which lies inside the critical hypersurface Z of M, the system is equivalent to
+the cotangent model (T ∗Tn)tw,(cc1,...,ccm) restricted to a neighbourhood of (T ∗Tn)0.
+Here c is the modular period of the connected component of Z containing T and the
+constants (c1, . . . , cm) are the invariants associated to the integrable system and its
+associated toric action.
+
+
+Part 3
+A KAM theorem for bm-symplectic
+manifolds
+
+The KAM theorem explains how integrable systems behave under small per-
+turbations.
+More precisely, it studies how an integrable system in action-angle
+coordinates responds to a small perturbation on its Hamiltonian. The trajectories
+of an integrable system in action-angle coordinates can be seen as linear trajectories
+over a torus. The KAM theorem finds a way to transform these original trajectories
+to other linear trajectories over some transformed torus. The KAM theorem states
+that most of these tori, and the linear solutions of the system on these tori, survive
+if the perturbation is small enough.
+In this part, we give a new KAM theorem for bm-symplectic manifolds with
+detailed proof. This is contained in the first chapter of this part. Moreover, we
+devote three more chapters to applications:
+(1) Desingularization of bm-integrable systems. We present a way to use
+the desingularization of bm-symplectic manifolds presented in [GMW17]
+to construct standard smooth integrable systems from bm-integrable sys-
+tems. This desingularized integrable system is uniquely defined.
+(2) Desingularization of the KAM theorem on bm-symplectic man-
+ifolds. In this section we use the desingularization of bm-integrable sys-
+tems in conjunction with the KAM theorem for bm-symplectic manifolds
+to deduce the original KAM theorem as well as a completely new KAM
+theorem for folded symplectic forms.
+(3) Potential applications to Celestial mechanics. We overview a list of
+motivating examples from Celestial mechanics where regularization trans-
+formations give rise to bm-symplectic forms. We discuss some potential
+applications of perturbation theory in this set-up.
+
+CHAPTER 6
+A new KAM theorem
+The objective of this chapter is to give a construction of KAM theory in the
+setting of bm-symplectic manifolds and with bm-integrable systems. The core of the
+chapter is the construction of the proper statement and the proof of the equivalent
+of the KAM theorem on bm-symplectic manifolds.
+This chapter is divided different sections:
+(1) On the structure of the proof. On this section we are going to present
+the main ideas that are going to appear in the proper statement and proof
+of the main theorem. The idea of the theorem is to build a sequence of
+bm-symplectomorphisms such that its limit transforms the hamiltonian to
+only depend on the action coordinates.
+(2) Technical results and definitions. On this section we present some
+technical results and definitions that are key for the proof of the main
+theorem.
+(3) KAM theorem on bm-symplectic manifolds.
+On this section we
+present the statement and the proof of the main result of this chap-
+ter.
+The proof is structured in 6 parts.
+In the first part we define
+the parameters that are going to be used to define the sequence of bm-
+symplectomorphisms. In the second part we build precisely this sequence
+of bm-symplectomorphisms. In the third part we see that the sequence of
+frequency maps of the transformed Hamiltonian functions at every step
+converges. In the fourth part we see that the sequence of bm-symplectomorphisms
+converges. In the fifth part we obtain results on the stability of the trajec-
+tories under the original perturbation. In the sixth part, we find bounds
+to explain how close the invariant tori are from the unperturbed. Finally,
+we obtain a bound for the measure of the set of invariant tori.
+6.1. On the structure of the proof
+The first thing we do is to reduce our study to the case the perturbation is not
+a bm-function but an analytic one. This is because any purely singular perturbation
+only affects the component in the direction of the modular vector field and can be
+easily controlled.
+The idea of the proof is really similar to the classical KAM case. We want
+to build a diffeomorphism such that its transformed hamiltonian only depends on
+the action coordinates. But it is not possible to build this diffeomorphism in one
+step. What we do, as it is done in the classical case, it is to build a sequence of
+diffeomorphisms such that the part of the hamiltonian depending on the angular
+variables decreases at every step. The idea is to remove the first K terms of its
+Fourier expression at every step while making K rapidly increase. This is done by
+43
+
+44
+6. A NEW KAM THEOREM
+assuming the diffeomorphism comes as the flow at time 1 generated by a Hamil-
+tonian function. In this way one can use the Lie Series in conjunction with the
+Fourier series to find the expression for the hamiltonian function that generates
+our diffeomorphism. The final diffeomorphism will be the composition of all the
+diffeomorphisms obtained at each step. One of the main difficulties of the proof, as
+in the classical case, is to prove that these diffeomorphisms converge and to prove
+some bounds of its norm.
+We also note that for our bm-symplectic setting, the diffeomorphisms we con-
+sider leave the defining function of the critical set invariant up to order m, this will
+have an important role later. Also observe in particular that the critical set can
+not be transformed by any perturbation given by a bm-function.
+Next we give some technical definitions and results. We define the norms we are
+going to use to do all the estimates. We set the notation for the proof and the state-
+ment of the theorem. We define the notion of non-resonance for a neighborhood
+of the critical set of the bm-symplectic manifold. We study the set of all possible
+non-resonant vectors. And we state the inductive lemma, which gives us estimates
+and constructions for every step of our sequence of diffeomorphisms.
+After all this discussion we are in conditions to properly state the bm-version of
+the KAM theorem. One important difference to the classical KAM theorem is that
+we have to guarantee that at Z the set of non-resonant vectors does not become
+the whole set of frequencies. This condition can be understood as the perturbation
+being smaller than some constant multiplied by the inverse of the modular period.
+The proof of the theorem is done in six different steps by following the structure
+on [DG96]. Since we are going to use the inductive lemma at every step, first we
+define the parameters and sets to which we are going apply such lemma. Then we
+check that we can actually apply the lemma and obtain some extra estimates for
+the results of the lemma. After this we see that the sequence of frequency vectors
+converges. We do the same with the sequence of canonical transformations. Then
+we get some bounds for the size of the components of the final diffeomorphism.
+Next we characterize the tori that survive by the perturbation. Finally we give
+some estimates for the measure of the set of these tori.
+Note that our version of the bm-KAM theorem improves the one in [KMS16a]
+in several ways. Firstly it is applicable to bm-symplectic structures not only for
+b-symplectic. Also we give several estimates that are not obtained in [KMS16a],
+this estimates have sense in a neighborhood of the critical set Z, while [KMS16a]
+only studied the behavior at Z. Finally the type of perturbation we consider is far
+more general, since we do not have any condition of the form of the perturbation
+but only on its size.
+6.1.1. Reducing the problem to an analytical perturbation. In the
+standard KAM, we assume to have an analytic Hamiltonian h(I) depending only
+on the action coordinates and we add to it a small analytical perturbation R(φ, I).
+This perturbed system receives the name of nearly integrable system. And then find
+a new coordinate system such that h(I) + R(φ, I) = ˜h(˜I) where most of the quasi-
+periodic orbits are preserved and can be mapped to the unperturbed quasi-periodic
+orbits by means of the coordinate change.
+In our setting we may assume h(I) to not be analytical and be a bm-function.
+Also the perturbation R(φ, I) may as well be considered a bm-function.
+In the
+
+6.1. ON THE STRUCTURE OF THE PROOF
+45
+following lines we justify without loss of generality that actually we can assume the
+perturbation to be analytical.
+Let us state this more precisely. Let (M, x, Z, ω, F) be a bm-manifold with a
+bm integrable system F on it. Consider action angle coordinates on a neighborhood
+of Z. Then we can assume the expressions:
+ω =
+
+
+m
+�
+j=1
+cj
+Ij
+1
+
+ dI1 ∧ dφ1 +
+n
+�
+i=2
+dIi ∧ dφi, and
+F = (q′
+0 log I1 +
+m−1
+�
+i=1
+q′
+i
+1
+Ii
+1
++ h(I), f2, . . . , fn)
+where h, f2, . . . , fn are analytical.
+Let the Hamiltonian function of our system be the first component of the
+moment map ˆh′ = q′
+0 log I1 + �m−1
+i=1 q′
+i
+1
+Ii
+1 + h = ζ′ + h, where ζ′ := q′
+0 log I1 +
+�m−1
+i=1 q′
+i
+1
+Ii
+1 . Note that dζ′ = �m
+i=1 ˆq′
+i
+1
+I′
+1 , where ˆq′
+i = −(i − 1)q′
+i−1. Note that by
+the result of the previous chapter cj/ˆq′
+j = K the modular period. In particular
+cm/ˆq′
+m = K.
+The hamiltonian system given by ˆh′ can be easily solved by φ = φ0 +u′t, I = I0
+where u′ is going to be defined in the following sections. Consider now a pertur-
+bation of this system:
+ˆH′ = ˆh′(I) = ˆR(I, φ), where ˆR is a bm-function ˆR(I, φ) =
+Rζ(I1)+R(I, φ) where Rζ(I1) = (r0 log I1 +�m−1
+i=1 ri 1
+Ii
+1 ) is the singular part. Then
+we can consider the perturbations Rζ(I1) and R(I, φ) separately. This way, we may
+consider Rζ(I) as part of ˆh′(I). Then we have a new hamiltonian
+ˆh(I) = (q′
+0 + r0) log I1 +
+m−1
+�
+i=1
+(q′
+i + ri) 1
+Ii
+1
++ h = q0 log I1 +
+m−1
+�
+i=1
+qi
+1
+Ii
+1
++ h.
+Now, instead of the identity Kˆq′
+j = cj we will have K(ˆqj − ˆrj) = cj, which
+implies K
+�
+1 −
+ˆrj
+ˆq′
+j+ˆrj
+�
+= cj
+ˆqj . In particular
+K
+�
+1 −
+ˆrm
+ˆq′m + ˆrm
+�
+= cm
+ˆqm
+Let us define K′ = K
+�
+1 −
+ˆrm
+ˆq′m+ˆrm
+�
+. So from now on we assume ˆh = q0 log I1 +
+�m−1
+i=1 qi 1
+Ii
+1 + h, that the perturbation R(φ, I) is analytical, and we have the condi-
+tion cm
+ˆqm = K′. Observe that this system with only the singular perturbation is still
+easy to solve in the same way that the system previous to this perturbation was.
+6.1.2. Looking for a bm-symplectomorphism. Assume we have a Hamil-
+tonian function H = ˆh(I) + R(φ, I) in action-angle coordinates. Where ˆh(I) is the
+singular component of the bm-integrable system, i.e.
+(6.1)
+ˆh(I) = h(I) + q0 log(I1) +
+m−1
+�
+i=1
+qi
+1
+Ii
+1
+,
+
+46
+6. A NEW KAM THEOREM
+where h(I) is analytical1. Assume also that the bm-symplectic form ω2 in these
+coordinates is expressed as:
+(6.2)
+ω =
+
+
+m
+�
+j=1
+cj
+Ij
+1
+
+ dI1 ∧ dφ1 +
+n
+�
+i=2
+dIi ∧ dφi.
+And finally, the expression for the frequency vector is:
+ˆu = ∂ˆh
+∂I =
+∂(h(I) + q0 log(I1) + �m−1
+i=1 qi 1
+Ii
+1 )
+∂I
+=
+�
+u1 +
+m
+�
+i=1
+ˆqi
+Ii
+1
+, u2, . . . , un
+�
+,
+where ˆq1 = q0 and ˆqi−1 = −iqi if i ̸= 0.
+The objective is to follow the steps of the usual KAM construction (the steps
+followed are highly inspired in [DG96]) replacing the standard symplectic form for
+ω and taking as hamiltonian the bm-function ˆh.
+Remark 6.1. The objective of the construction is to find a diffeomorphism
+(actually a bm-symplectomorphism) ψ such that H ◦ ψ = h(˜I). This is done induc-
+tively, by taking H ◦ ψ = H ◦ φ1 ◦ . . .◦ φq ◦ . . ., while trying to make R(φ, I) smaller
+at every step.
+Let us focus in one single step
+Recall the classical formula:
+Lemma 6.2. See [DG96].
+f ◦ φt =
+∞
+�
+j=0
+tj
+j!Lj
+W f,
+Lj
+W f = {Lj−1
+W f, W}
+Where W is the Hamiltonian that generates the flow φt, and {·, ·} is the correspond-
+ing Poisson bracket.
+We will denote rk(H, W, t) = �∞
+j=k
+tj
+j! Lj
+W H.
+1If another component of the moment map is chosen to be the hamiltonian of the system,
+the result still holds: the computations can be replicated assuming ˆh(I) = h(I).
+2In classical KAM, ω is used to denote the frequency vector ∂h
+∂I . We need ω to denote the
+bm-symplectic form so we are going to use u to denote the frequency vector.
+
+6.1. ON THE STRUCTURE OF THE PROOF
+47
+(6.3)
+H ◦ φ = H ◦ φ|t=1
+=
+∞
+�
+j=0
+tj
+j! Lj
+W H
+����
+ˆh+R
+������
+t=1
+=
+ˆh + R{ˆh + R, W} + r2(H, W, 1)
+=
+ˆh + R + {ˆh, W} + {R, W} + r2(ˆh, W, 1)
++r2(R, W, 1)
+=
+ˆh +
+R + {ˆh, W}
+�
+��
+�
+We want to cancel
+this term as
+fast as we can
++r2(ˆh, W, 1) + r2(R, W, 1)
+We want {ˆh, W} +R≤k = 0, equivalently {W, ˆh} = R≤k, where R≤k means the
+Fourier expression of R up to order K:
+R≤k =
+�
+k∈Rn
+|k|1≤K
+Rk(I)eik·φ
+Let us impose the condition {W, ˆh} = R≤K. Let us write the expression of the
+Poisson bracket associated to the bm-symplectic form.
+{W, ˆh}
+=
+�
+1
+�m
+j=1
+cj
+Ij
+1
+� �
+∂W
+∂φ1
+∂ˆh
+∂I1
+− ∂W
+∂I1
+∂ˆh
+∂φ1
+�
++
+n
+�
+i=2
+�
+∂W
+∂φi
+∂ˆh
+∂Ii
+− ∂W
+∂Ii
+∂ˆh
+∂φi
+�
+Because ˆh depends only on I,
+∂ˆh
+∂φi = 0 for all i. Moreover, the singular part of
+the bm-function only depends on I1 and hence its derivatives with respect to the
+other variables are also 0. Using that ∂ˆh
+∂I = u + �m
+i=1
+ˆqi
+Ii
+1 the previous expression
+can be simplified:
+{W, ˆh}
+=
+
+u1 + �m
+i=1
+ˆqi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+
+ ∂W
+∂φ1
++
+n
+�
+i=2
+∂W
+∂φi
+ui
+To expand the expression further we develop W in its Fourier expression: W =
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφ. The Fourier expansion is added up to order K, because it is
+only necessary for the expressions to agree up to order K. With this notations the
+condition becomes:
+{W, ˆh}≤K
+=
+
+u1 + �m
+i=1
+ˆqi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+
+
+∂
+∂φ1
+
+
+
+
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφ
+
+
+
+
+
+48
+6. A NEW KAM THEOREM
++
+n
+�
+j=2
+uj
+∂
+∂φj
+
+
+
+
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφ
+
+
+
+
+=
+
+u1 + �m
+i=1
+ˆqi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+
+
+
+
+
+
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφik1
+
+
+
+
++
+n
+�
+j=2
+uj
+
+
+
+
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφikj
+
+
+
+
+=
+�
+k∈Rn
+|k|1≤K
+Wk(I)eikφ ·
+
+ik1
+
+u1 + �m
+i=1
+ˆqi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+
+ +
+n
+�
+j=2
+ikjuj
+
+
+= R≤K
+Then, it is possible to make the two sides of the equation equal by imposing
+the condition term by term:
+(6.4)
+Wk(I)
+=
+Rk(I)
+1
+i
+�
+k1
+�
+u1+�m
+i=1
+ˆ
+qi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+�
++ �n
+j=2 kjuj
+�
+=
+Rk(I)
+1
+i
+�
+k1
+�
+u1+�m
+i=1
+ˆ
+qi
+Ii
+1
+�m
+j=1
+cj
+Ij
+1
+�
++ ¯k¯u
+�,
+where we adopted the notation �n
+j=2 kjuj = ¯k¯u.
+Remark 6.3. Observe that the expression 6.4 has no sense when k = ⃗0 and
+hence {W, h}0 = R03 can not be solved. Let W0(I) = 0, then {h, W}≤K = R≤K −
+R0.
+Plugging the results above into the equation 6.3, one obtains:
+H ◦ φ = ˆh + R0 + R≥K + r2(ˆh, W, 1) + r1(R, W, 1)
+With this construction the diffeomorphism φ is found. But this is only the
+first of many steps. If q denotes the number of the iteration of this procedure, in
+general, we obtain:
+3The zero term of the Fourier series can be seen as the angular average of the function
+
+6.1. ON THE STRUCTURE OF THE PROOF
+49
+(6.5)
+H(q) = H(q−1) �� φ(q)
+=
+ˆh(q−1) + R(q−1)
+0
++ R(q−1)
+≥K
++r2(h(q−1), W (q), 1) + r1(R(q−1), W (q), 1),
+and at every step:
+(6.6)
+�ˆh(q) = ˆh(q−1) + R(q−1)
+0
+R(q) = R(q−1)
+>K
++ r2(ˆh(q−1), W (q), 1) + r1(R(q−1), W (q), 1)
+6.1.3. On the change of the defining function under
+bm-symplectomorphisms. Note that since we are considering bm-manifolds it
+only makes sense to consider I1 up to order m, see [Sco16]. When talking about
+defining functions we are interested in [I1], its jet up to order m. By definition bm-
+maps preserve I1 up to order m and bm-vector fields X are such that LX(I1) = g·Im
+1
+for g ∈ C∞(M).
+Lemma 6.4. Let φt be the integral flow of X a bm-vector field, then φt is a
+bm-map.
+Proof. We want
+I1 ◦ φt = I1 + Im
+1 · g
+for some g ∈ C∞(M). We will use 6.2.
+I1 ◦ φt =
+∞
+�
+j=0
+tj
+j!Lj
+XI1 = I1 + LX(I1) +
+∞
+�
+j=2
+tj
+j!Lj
+XI1
+= I1 + Im
+1 +
+∞
+�
+j=2
+tj
+j!Lj
+XI1.
+On the other hand, let us prove by induction Lk
+XI1 = g(k)Im
+1 . The first case is
+obvious, assume the case k holds and let us prove the case k + 1.
+Lk+1
+X
+I1
+=
+{Lk
+XI1, X}
+=
+{g(k)Im
+1 , X}
+=
+(LXg(q))Im
+1 + g(k) · mIm−1
+1
+LXI1
+=
+(LXg(k) + g(k) · m · Im−1
+1
+· g)Im
+1
+=
+g(k+1)Im
+1
+where g(k+1) = LXg(k) + g(k) · m · Im−1
+1
+· g.
+□
+Lemma 6.5. The Hamiltonian vector flow of some smooth hamiltonian function
+h is a bm-vector field.
+Proof. At each point of Z the following identity holds LXhI1 = Im
+1
+∂f
+∂φ1 . The
+result can be extended at a neighborhood of Z.
+□
+Observe that combining the two previous results we get that the hamiltonian
+flow of a function preserves I1 up to order m.
+
+50
+6. A NEW KAM THEOREM
+6.2. Technical results
+As the non-singular part of our functions we will be considering analytic func-
+tions on T × G, G ⊂ Rn.
+The easiest way to work with these functions is to
+consider them as holomorphic functions on some complex neighborhood. Let us
+define formally this neighborhood.
+Wρ1(Tn) := {φ : ℜφ ∈ Tn, |ℑφ|∞ ≤ ρ1},
+Vρ2(G) := {I ∈ Cn : |I − I′| ≤ ρ2 for some I′ ∈ G},
+Dρ(G) := Wρ1(Tn) × Vρ2(G),
+where | · |∞ denotes the maximum norm and | · |2 denotes de Euclidean norm.
+Now it is necessary to clarify the norms that are going to be used on these sets.
+Definition 6.6. Let f be an action function (only depending on the I-coordinates),
+and F an action vector field.
+|f|G,η := supI∈Vη(G) |f(I)|,
+|f|G := |f|G,0
+|F|G,η,p := supI∈Vη(G) |F(I)|p,
+|F|G,η := |F|G,η,2
+Now, assume f(I, φ) to be an action-angle function written using its Fourier ex-
+pansion as �
+k∈Zn fk(I)eik·φ, and F to be an action-angle vector field.
+|f|G,ρ := sup(φ,I)∈Dρ(G) |f(I)|,
+∥f∥G,ρ := �
+k∈Zn |fk|G,ρ2e|k|1ρ1
+|F|G,ρ,p := �
+k∈Zn |Fk|G,ρ2,pe|k|1ρ1,
+∥F∥G,ρ = ∥F∥G,ρ,2
+Lemma 6.7 (Cauchy Inequality).
+����
+∂f
+∂φ
+����
+G,(ρ1−δ1,ρ2),1
+≤
+1
+eδ1
+∥f∥G,ρ
+����
+∂f
+∂I
+����
+G,(ρ1,ρ2−δ2),∞
+≤ 1
+δ2
+∥f∥G,ρ
+Definition 6.8. If Df = ( ∂f
+∂φ, ∂f
+∂I ),
+∥Df∥G,ρ,c := max
+�
+∥∂f
+∂φ∥G,ρ,1, c∥∂f
+∂I ∥G,ρ,∞
+�
+Definition 6.9. To simplify our notation, let us define:
+A(I1) =
+�m
+j=1
+ˆqj
+Ij
+1
+�m
+j=1
+cj
+Ij
+1
+and
+B(I1) =
+1
+�m
+j=1
+cj
+Ij
+1
+.
+Remark 6.10. With this notation, equation 6.4 can be written as:
+Wk(I) =
+Rk(I)
+i(k1B(I1)u1 + ¯k¯u + k1A(I1))
+Observe that A(I1) and B(I1) are analytic (holomorphic on the complex ex-
+tended domain) where the denominator does not vanish. We can assume that this
+does not happen by shrinking the domain G in the direction of I1. Observe, in
+particular, that when I1 → 0, A(I1) → ˆqm/cm = 1/K′ the inverse of the modular
+period and B(I1) → 0. In this way, the norms of A(I1) and B(I1) are bounded and
+well defined. We will denote these norms by KA and KB respectively. Also, since
+
+6.2. TECHNICAL RESULTS
+51
+A(I1) and B(I1) are analytic, their derivatives will also be bounded, and we will
+denote the norms of these derivatives by KA′ and KB′.
+To further simplify the notation in the following computations we introduce
+the definition:
+Definition 6.11.
+¯
+A =
+�
+A
+0
+�
+and
+¯B =
+�
+B
+0
+0
+Idn−1,n−1
+�
+Remark 6.12. With this notation, equation 6.4 can be written as:
+(6.7)
+Wk(I) =
+Rk(I)
+i(k ¯B(I1)u + k ¯
+A(I1))
+Definition 6.13. Having fixed ω, a bm-symplectic form (as in equation 6.2)
+and ˆh a bm-function (as in equation 6.1) as a hamiltonian. Given an integer K and
+α > 0, F ⊂ Rn (or Cn) the space of frequencies is said to be α, K-non-resonant
+with respect to (c1, . . . , cm) and (ˆq1, . . . , ˆqm) if
+|k ¯B(I1)u + k ¯
+A(I1)| ≥ α, ∀k ∈ Z \ {0}, |k|1 ≤ K, ∀u ∈ F.
+We are going to use the notation α, K, c, ˆq-non-resonant.
+Remark 6.14. The non-resonance condition is established on u = ∂h/∂I, not
+on ˆu = ∂ˆh/∂I, because our non-resonance condition already takes into account the
+singularities. In this way we can use the analytic character of u.
+Remark 6.15. If
+�� ∂u
+∂I
+��
+G,ρ2 is bounded by M ′, then
+�� ∂
+∂I
+� ¯Bu + ¯
+A
+���
+G,ρ2 is also
+bounded:
+(6.8)
+�� ∂
+∂I
+� ¯Bu + ¯
+A
+���
+G,ρ2
+≤
+��� ∂ ¯
+B
+∂I u + ¯B ∂u
+∂I + ∂ ¯
+A
+∂I
+���
+G,ρ2
+≤
+KB′|u|G,ρ2 + KBM ′ + KA =: M.
+Remark 6.16. When we consider the standard KAM theorem, the frequency
+vector u is relevant because the solution to the Hamilton equations of the unper-
+turbed problem has the form:
+I = I0,
+φ = φ0 + ut.
+Let us see what plays the role of u in our bm-KAM theorem. Let us find the
+coordinate expression of the solution to ιXˆhω = dˆh, where ω is a bm-symplectic
+form in action-angle coordinates.
+Xˆh = ˙I1
+∂
+∂I1
++ . . . + ˙In
+∂
+∂In
+,
+where ˙I1, . . . , ˙In are the functions we want to find.
+dˆh =
+
+
+m
+�
+j=1
+ˆqi
+1
+Ij
+1
+
+ dI1 + dh,
+and hence,
+
+52
+6. A NEW KAM THEOREM
+Xˆh = Π(dˆh, ·) =
+�m
+i=1
+ˆqi
+Ii
+1
+�m
+i=1
+cj
+Ij
+1
+∂
+∂φi
++ Xh.
+Hence φ = φ0 + ( ¯Bu + ¯
+A
+� �� �
+u′
+)t. So the frequency vector that we are going to be
+concerned about is going to be u′ instead of ˆu =
+∂
+∂I ˆh.
+Lemma 6.17. If u is one-to-one from G to its image then u′ = ¯Bu + ¯
+A is
+also one-to-one from G′ to its image in a neighborhood of Z, while at Z it is the
+projection of u such that the first coordinate is sent to ˆqm
+cm = 1/K′ the inverse of the
+modular period, were G′ ⊆ G.
+Proof. Because
+u′ =
+
+
+1
+�m
+j=1
+cj
+Ij
+1
+u1 +
+�m
+j=1
+ˆqj
+Ij
+1
+�m
+j=1
+cj
+I1
+, u2, . . . , un
+
+ ,
+and B is invertible outside I1 = 0, shrinking G if necessary in the first dimension
+the map is one-to-one. But at the critical set {I1 = 0}, u′ is a projection of u where
+the first component is sent to the constant value ˆqm
+cm =
+1
+K′ .
+□
+Lemma 6.18. If u(G) is α, K, c, ˆq-non-resonant, then u(Vρ2(G)) is α
+2 , K, c, ˆq-
+non-resonant, assuming that ρ2 ≤
+α
+2MK and
+�� ∂u
+∂I
+��
+G,ρ2 ≤ M ′
+Proof. Fix k ∈ Z \ {0}, we want to bound |k ¯B(I1)v + k ¯
+A(I1)| where v ∈
+u(Vρ2(G)) as a function on v. Given v ∈ u(Vρ2(G)) we ask whether there is any
+bound for the distance to some v′ ∈ u(G).
+v ∈ u(Vρ2(G)) ⇒ v = u(x), x ∈ Vρ2(G) ⇒ ∃y ∈ G such that |x − y| ≤ ρ2.
+Take v′ = u(y).
+|v − v′| ≤ |x − y|
+����
+∂u
+∂I
+����
+G,ρ2
+≤ ρ2M ′ ≤ ρ2M/KB ≤
+α
+2MK M/KB =
+α
+2KKB
+.
+Where we used equation 6.8 in the third inequality.
+|k1B(I1)v1 + ¯k¯v + k1A(I1)|
+≥
+|k1B(I1)v′
+1 + ¯k ¯v′ + k1A(I1)|
+�
+��
+�
+≥α
+−|k1B(I1)(v1 − v′
+1) + ¯k(¯v − ¯v′)|
+≥
+α − KB |k · (v − v′)|
+�
+��
+�
+≤Kα/(2KKB)
+≥
+α − α/2 = α/2
+□
+Proposition 6.19. Let ˆh(I) be a bm-function as in equation 6.1. Assume h(I)
+and R(φ, I) be real analytic on Dρ(G), u(G) = ∂h
+∂I (G) is α, K, c, ˆq-non-resonant.
+Assume also that | ∂
+∂I u|G,ρ2 ≤ M ′ and ρ2 ≤
+α
+2MK . Let c > 0 given. Then R0(φ, I),
+
+6.2. TECHNICAL RESULTS
+53
+W≤K(φ, I) given by the previous construction are both real analytic on Dρ(G) and
+the following bounds hold
+(1) ||DR0||G,ρ,c ≤ ||DR||G,ρ,c
+(2) ||D(R − R0)||G,ρ,c ≤ ||DR0||G,ρ,c
+(3) ||DW||G,ρ,c ≤ 2A
+α ||DR0||G,ρ,c
+Where A = 1 + 2Mc
+α
+Proof. Inequalities 1 and 2 are obvious because of the Fourier expression.
+Let us prove inequality 3.
+Let us expand R(φ, I) and W(φ, I) in their Fourier
+expression:
+R =
+�
+k∈Rn
+Rk(I)eik·φ,
+W =
+�
+k∈Rn
+Wk(I)eik·φ.
+We will bound this expression finding term-by-term bounds.
+∂R
+∂φ =
+�
+k∈Rn
+Rk(I)eik·φik.
+Hence, if we denote [ ∂R
+∂φ ]k the k-th term of the Fourier expansion of ∂R
+∂φ , we
+have:
+�∂R
+∂φ
+�
+k
+= Rkik.
+Let us compute the derivative of Wk with respect to the I variables:
+∂Wk
+∂I
+=
+∂
+∂I
+�
+Rk
+i(k ¯B(I1)u + k ¯
+A(I1))
+�
+=
+∂Rk/∂I
+i(k ¯B(I1)u + k ¯
+A(I1))) − Rki ∂
+∂I (k ¯B(I1)u + k ¯
+A(I1)))
+[i(k ¯B(I1)u + k ¯
+A(I1)))]2
+=
+∂Rk/∂I
+i(k ¯B(I1)u + k ¯
+A(I1))) + Rkik ∂
+∂I ( ¯B(I1)u + ¯
+A(I1)))
+[(k ¯B(I1)u + k ¯
+A(I1)))]2
+=
+∂Rk/∂I
+i(k ¯B(I1)u + k ¯
+A(I1))) +
+[ ∂Rk
+∂φ ]k ∂
+∂I ( ¯B(I1)u + ¯
+A(I1)))
+[(k ¯B(I1)u + k ¯
+A(I1)))]2
+.
+Then, we take norms (| · |G,ρ2,∞) on each side of the equation.
+����
+∂Wk
+∂I
+����
+G,ρ2,∞
+≤
+2
+α
+����
+∂Rk
+∂I
+����
+G,ρ2,∞
++ 4M
+α2
+����
+�∂Rk
+∂φ
+�
+k
+����
+G,ρ2,∞
+≤
+2
+α
+����
+∂Rk
+∂I
+����
+G,ρ2,∞
++ 4M
+α2
+����
+�∂Rk
+∂φ
+�
+k
+����
+G,ρ2,1
+.
+Taking the supremum at the whole domain:
+����
+∂Wk
+∂I
+����
+G,ρ2,∞
+≤
+2
+α
+����
+∂Rk
+∂I
+����
+G,ρ2,∞
++ 4M
+α2
+����
+�∂Rk
+∂φ
+�
+k
+����
+G,ρ2,1
+.
+Moreover,
+
+54
+6. A NEW KAM THEOREM
+∂W(I)
+∂φ
+=
+∂
+∂φ
+� �
+k∈Rn
+Wk(I)eik·φ
+�
+=
+∂
+∂φ
+� �
+k∈Rn
+ikWk(I)eik·φ
+�
+.
+Hence, the k-th term of the Fourier series of ∂W
+∂φ is
+�∂W
+∂φ
+�
+k
+= Wkik =
+Rk
+i(k ¯B(I1)u + k ¯
+A(I1)))ik
+=
+1
+i(k ¯B(I1)u + k ¯
+A(I1)))
+�∂R
+∂φ
+�
+k
+.
+Taking norms (∥ · ∥G,ρ,1) at each side:
+����
+∂W
+∂φ
+����
+G,ρ,1
+≤ 2
+α
+����
+∂W
+∂φ
+����
+G,ρ,1
+.
+Then,
+∥DW∥G,ρ,c
+=
+max
+�����
+∂W
+∂φ
+����
+G,ρ,1
+, c
+����
+∂W
+∂I
+����
+G,ρ,∞
+�
+≤
+max
+�
+2
+α
+����
+∂R
+∂φ
+����
+G,ρ,1
+, c 2
+α
+����
+∂R
+∂I
+����
+G,ρ2,∞
++ c4M
+α2
+����
+∂R
+∂φ
+����
+G,ρ2,1
+�
+≤
+max
+�
+2
+α
+����
+∂R
+∂φ
+����
+G,ρ,1
+, 2
+α ∥DR∥G,ρ2,c + c4M
+α2 ∥DR∥G,ρ2,c
+�
+=
+max
+�
+2
+α
+����
+∂R
+∂φ
+����
+G,ρ,1
+, 2
+α
+�
+1 + 2M
+α c
+�
+∥DR∥G,ρ2,c
+�
+≤
+2
+α
+�
+1 + 2M
+α c
+�
+∥DR∥G,ρ2,c
+≤
+2
+αA ∥DR∥G,ρ2,c,
+where A is as desired.
+□
+Recall the Cauchy inequalities, see [P¨os93]:
+(6.9)
+����
+∂f
+∂φ
+����
+G,(ρ1,ρ2),1
+≤
+1
+eδ1
+∥f∥G,ρ
+����
+∂f
+∂I
+����
+G,(ρ1,ρ2−δ2),∞
+≤
+1
+δ2
+∥f∥G,ρ
+
+6.2. TECHNICAL RESULTS
+55
+Lemma 6.20. Let f, g be analytic functions on Dρ(G), where 0 < δ = (δ1, δ2) <
+ρ = (ρ1, ρ2) and c > 0. Define ˆδc := min(cδ1, δ2). The following inequalities hold:
+(1) ∥Df∥G,ρ−δ,c ≤
+c
+ˆδc ∥f∥G,ρ
+(2) ∥{f, g}∥G,ρ ≤ 2
+c∥Df∥G,ρ,c · ∥Dg∥G,ρ,c
+(3) ∥D(f>K)∥G,(ρ−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c
+Proof. Let us prove each point separately.
+(1) Using the Cauchy inequalities one obtains the following:
+����
+∂f
+∂φ
+����
+G,ρ−δ,1
+=
+����
+∂f
+∂φ
+����
+G,(ρ1−δ1,ρ2−δ2),1
+≤
+����
+∂f
+∂φ
+����
+G,(ρ1−δ1,ρ2),1
+≤
+1
+eδ1
+∥f∥G,ρ,
+����
+∂f
+∂I
+����
+G,ρ−δ,∞
+=
+����
+∂f
+∂I
+����
+G,(ρ1−δ1,ρ2−δ2),∞
+≤
+����
+∂f
+∂I
+����
+G,(ρ1,ρ2−δ2),∞
+≤ 1
+δ1
+∥f∥G,ρ.
+Putting the two inequalities inside the definition of the norm:
+∥Df∥G,ρ−δ,c
+=
+max
+�����
+∂f
+∂φ
+����
+G,ρ−δ,1
+, c
+����
+∂f
+∂I
+����
+G,ρ−δ,∞
+�
+≤
+max
+� 1
+eδ1
+∥f∥G,ρ, c
+δ2
+∥f∥G,ρ
+�
+≤
+max
+� 1
+eδ1
+c
+c, c
+δ2
+�
+∥f∥G,ρ
+≤
+max
+� c
+eˆδc
+, c
+ˆδc
+�
+∥f∥G,ρ,
+where the last inequality holds because ˆδc = min(cδ1, δ2).
+(2) Let us find the expression of {f, g} for a bm-symplectic structure. {f, g} =
+ω(Xf, Xg) where Xf and Xg are such that ιXf ω = df and ιXgω = dg.
+Let restrict the computations only to f.
+df =
+n
+�
+i=1
+∂f
+∂φ1
+dφ1,
+Xf =
+n
+�
+i=1
+ai
+∂
+∂φi
++
+n
+�
+i=1
+bi
+∂
+∂φi
+.
+Where ai and bi are coefficients to be determined by imposing the
+following condition:
+ιXf ω =
+
+
+m
+�
+j=1
+cj
+Ij
+1
+
+ (a1dI1 − b1dφ1) +
+n
+�
+i=2
+(aidIi − bidφi) = df.
+Then, solving for the coefficients the following expressions are ob-
+tained:
+
+56
+6. A NEW KAM THEOREM
+a1 =
+1
+��m
+j=1
+cj
+Ij
+1
+� ∂f
+∂φ1
+and
+ai = ∂f
+∂φi
+for i ̸= 1,
+b1 = −
+1
+��m
+j=1
+cj
+Ij
+1
+� ∂f
+∂φ1
+and
+bi = − ∂f
+∂φi
+for i ̸= 1.
+Hence, the expression for the hamiltonian vector fields becomes:
+Xf =
+1
+��m
+j=1
+cj
+Ij
+1
+�
+� ∂f
+∂φ1
+∂
+∂φ1
+− ∂f
+∂I1
+∂
+∂I1
+�
++
+n
+�
+i=1
+� ∂f
+∂φi
+∂
+∂φi
+− ∂f
+∂Ii
+∂
+∂Ii
+�
+,
+Xg =
+1
+��m
+j=1
+cj
+Ij
+1
+�
+� ∂g
+∂φ1
+∂
+∂φ1
+− ∂g
+∂I1
+∂
+∂I1
+�
++
+n
+�
+i=1
+� ∂g
+∂φi
+∂
+∂φi
+− ∂g
+∂Ii
+∂
+∂Ii
+�
+.
+Then the Poisson bracket applied to the two functions:
+{f, g} = ω(Xf, Xg)
+=
+1
+��m
+j=1
+cj
+Ij
+1
+�
+� ∂f
+∂I1
+∂g
+∂φ1
+− ∂f
+∂φ1
+∂g
+∂I1
+�
++
+n
+�
+i=2
+� ∂f
+∂Ii
+∂g
+∂φi
+− ∂f
+∂φi
+∂g
+∂Ii
+�
+.
+And hence the norm of the Poisson bracket becomes:
+∥{f, g}∥G,ρ
+=
+�������
+1
+��m
+j=1
+cj
+Ij
+1
+�
+� ∂f
+∂I1
+∂g
+∂φ1
+− ∂f
+∂φ1
+∂g
+∂I1
+�
++
+n
+�
+i=2
+� ∂f
+∂Ii
+∂g
+∂φi
+− ∂f
+∂φi
+∂g
+∂Ii
+������
+G,ρ
+≤
+�����
+n
+�
+i=1
+� ∂f
+∂Ii
+∂g
+∂φi
+− ∂f
+∂φi
+∂g
+∂Ii
+������
+G,ρ
+Where we assumed
+����m
+j=1
+cj
+Ij
+1
+��� ≥ 1. This assumption makes sense,
+because we are interested in the behaviour close the critical set Z. Close
+enough to the critical set this expression holds. Then,
+∥{f, g}∥G,ρ
+≤
+n
+�
+i=1
+����
+∂f
+∂Ii
+����
+G,ρ
+����
+∂g
+∂��i
+����
+G,ρ
++
+n
+�
+i=1
+����
+∂f
+∂φi
+����
+G,ρ
+����
+∂g
+∂Ii
+����
+G,ρ
+≤
+����
+∂f
+∂I
+����
+G,ρ,∞
+����
+∂g
+∂I
+����
+G,ρ,1
++
+����
+∂f
+∂I
+����
+G,ρ,1
+����
+∂g
+∂I
+����
+G,ρ,∞
+≤
+1
+c |Df∥G,ρ,c∥Dg∥G,ρ,c + 1
+c|Df∥G,ρ,c∥Dg∥G,ρ,c
+
+6.2. TECHNICAL RESULTS
+57
+≤
+2
+c ∥Df∥G,ρ,c∥Dg∥G,ρ,c.
+(3) Lastly,
+∥D(f>K)∥G,(ρ1−δ1,ρ2),1
+= max
+�����
+∂f>K
+∂φ
+����
+G,(ρ1−δ1,ρ1),1
+, c
+����
+∂f>K
+∂I
+����
+G,(ρ1−δ1,ρ1),∞
+�
+.
+We will proceed by bounding each term separately. On one hand:
+����
+∂f
+∂φ
+����
+G,(ρ1,ρ2),1
+=
+�����
+�
+k∈Zn
+ikfk(I)eikφ
+�����
+G,(ρ1,ρ2),1
+≥
+�
+k∈Zn
+k ∥fk(I)∥G,ρ2,1 e|k|1ρ1
+≥
+�
+k∈Zn
+|k|1>K
+k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1+δ1−δ1)
+≥
+eKδ1
+�
+k∈Zn
+|k|1>K
+k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1−δ1)
+=
+eKδ1
+����
+∂f>K
+∂φ
+����
+G,(ρ1−δ1,ρ2),1
+.
+On the other hand:
+����
+∂f
+∂I
+����
+G,(ρ1,ρ2),∞
+=
+�����
+�
+k∈Zn
+∂fk(I)
+∂I
+eikφ
+�����
+G,(ρ1,ρ2),∞
+≥
+�
+k∈Zn
+����
+∂fk(I)
+∂I
+����
+G,ρ2,∞
+e|k|1ρ1
+≥
+�
+k∈Zn
+|k|1>K
+����
+∂fk(I)
+∂I
+����
+G,ρ2,∞
+e|k|1(ρ1+δ1−δ1)
+≥
+eKδ1
+�
+k∈Zn
+|k|1>K
+����
+∂fk(I)
+∂I
+����
+G,ρ2,∞
+e|k|1(ρ1−δ1)
+≥
+eKδ1
+����
+∂f>K
+∂I
+����
+G,(ρ1−δ1,ρ2),∞
+.
+Hence ∥D(f>k)∥G,(ρ1−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c.
+
+58
+6. A NEW KAM THEOREM
+□
+Now we define a norm that indicates how close a map Φ is to the identity.
+Definition 6.21. Let x = (φ, I) ∈ C2n, then
+|x|c := max(|φ|1, c|I|∞)
+Definition 6.22. For a map Υ : Dρ(G) → C2n its norm and the norm of its
+derivative its defined as:
+|Υ|G,ρ,c :=
+sup
+x∈Dρ(G)
+|Υ(x)|c,
+|DΥ|G,ρ,c :=
+sup
+x∈Dρ(G)
+|DΥ(x)|c,
+where |DΥ(x)|c = sup
+y∈R2n
+|y|c=1
+|DΥ(x) · y|c
+Lemma 6.23. If Υ is analytic on Dρ(G), then |DΥ|G,ρ−δ,C ≤ |Υ|G,ρ,c
+ˆδc
+Proof. Observe that if we consider ∥.∥ any norm on Cn and a matrix A of
+size n × n, and ∥A∥ defines the induced norm of matrices i.e.
+∥A∥ = sup
+y∈C2n
+∥y∥=1
+∥A · y∥
+then one has that ∥(∥a1∥′, . . . , ∥an∥′)∥ ≤ ∥A∥ where aj denotes the j-th row of A.
+Also note that ∥ · ∥′ can be a any norm consider the infinity norm. This can be
+easily proven in the following way:
+∥A · y∥ =
+�������
+
+
+
+a1 · y
+...
+an · y
+
+
+
+�������
+≤
+�������
+
+
+
+∥a1∥′∥y∥′
+...
+∥an∥′∥y∥′
+
+
+
+�������
+Where ∀y ∈ Cn such that ∥y∥ = 1. Let aj be the rows of DΥ(x),
+aj =
+�∂Υj
+∂φ , ∂Υj
+∂I
+�
+,
+and be ∥aj∥′ its norm. With this property in mind we proceed as follows:
+|DΥ|G,ρ−δ,c
+=
+sup
+x∈Dρ−δ(G)
+|DΥ(x)|c
+≤
+sup
+x∈Dρ−δ(G)
+|(|a1|∞, . . . , |an|∞)|c
+≤
+���
+�
+supx∈Dρ−δ ∥DΥ1∥∞ , . . . , supx∈Dρ−δ ∥DΥ2n∥∞
+����
+c
+=
+���
+�
+∥DΥ1∥G,ρ−δ,∞ , . . . , ∥DΥ2n∥G,ρ−δ,∞
+����
+c
+≤
+���
+�
+1
+δ1 ∥Υ1∥G,ρ , . . . , 1
+δ1 ∥Υ2n∥G,ρ
+����
+c
+
+6.2. TECHNICAL RESULTS
+59
+≤
+1
+ˆδc
+���∥Υ1∥G,ρ , . . . , ∥Υ2n∥G,ρ
+���
+c
+=
+1
+ˆδc supx∈Dρ(G) |Υ1, . . . , Υ2n|c =
+1
+ˆδc supx∈Dρ(G) |Υ|c
+=
+1
+ˆδc |Υ|G,ρ,c
+□
+Lemma 6.24. Let W be an analytic function on Dρ(G), ρ > 0 and let Φt be its
+Hamiltonian flow at time t (t > 0). Let δ = (δ1, δ2) > 0 and c > 0 given. Assume
+that ∥DW∥G,ρ,c ≤ ˆδc. Then, Φt maps Dρ−tδ(G) into Dρ(G) and one has:
+(1) |Φt − Id|G,ρ−tδ,c ≤ t∥DW∥G,ρ,c,
+(2) Φ(Dρ(G)) ⊃ Dρ−tδ(G) for ρ′ ≤ ρ − tδ,
+(3) Assuming that ∥DW∥G,ρ,c < ˆδc/2e, for any given function f analytic on
+Dρ(G), and for any integer m ≥ 0, the following bound holds:
+∥rm(f, W, t)∥G,ρ−tδ
+≤
+∞
+�
+l=0
+�
+1
+�l+m
+m
+� ·
+�2e∥DW∥G,ρ,c
+ˆδc
+�l�
+tm
+m!∥Lm
+Wf∥G,ρ
+= γm
+�2e∥DW∥G,ρ,c
+ˆδc
+�
+· tm∥Lm
+Wf∥G,ρ,
+where for 0 ≤ x ≤ 1 we define
+γm(x) :=
+∞
+�
+l=0
+l!
+(l + m)!xl
+Proof. During the proof we are going to denote Φs(φ0, I0) by (φ(s), I(s)).
+Let us find the coordinate expression of the hamiltonian flow for the expression
+6.2 of a bm-symplectic form. Recall that the equation for the hamiltonian flow is
+d
+dsφi(s) = {φi, W} and
+d
+dsIi(s) = {Ii, W}.
+{φi, W} =
+1
+��m
+j=1
+cj
+Ij
+1
+�
+�∂φi
+∂I1
+· ∂W
+∂φ1
+− ∂φi
+∂φ1
+· ∂W
+∂I1
+�
++
+n
+�
+j=2
+�∂φi
+∂Ij
+· ∂W
+∂φj
+− ∂φi
+∂φj
+· ∂W
+∂Ij
+�
+.
+Hence,
+d
+dsφi(s) = −
+1
+��m
+j=1
+cj
+Ij
+1
+� ∂W
+∂I1
+if i = 1 and d
+dsφi(s) = −∂W
+∂Ii
+if i ̸= 1.
+On the other side,
+{Ii, W} =
+1
+��m
+j=1
+cj
+Ij
+1
+�
+� ∂Ii
+∂I1
+· ∂W
+∂φ1
+− ∂Ii
+∂φ1
+· ∂W
+∂I1
+�
++
+n
+�
+j=2
+� ∂Ii
+∂Ij
+· ∂W
+∂φj
+− ∂Ii
+∂φj
+· ∂W
+∂Ij
+�
+.
+
+60
+6. A NEW KAM THEOREM
+Hence,
+d
+dsIi(s) =
+1
+��m
+j=1
+cj
+Ij
+1
+� ∂W
+∂φ1
+if i = 1 and d
+dsIi(s) = ∂W
+∂φi
+if i ̸= 1.
+(1) Assume now that 0 < s0 ≤ t. Then,
+|φ(s0) − φ0|∞
+≤
+s0 sup0<s≤s0 |φ′(s)|∞
+=
+s0 sup0<s≤s0 (max(|φ′
+1(s)|, . . . , |φ′
+n(s)|))
+=
+s0
+sup
+0<s≤s0
+
+
+max
+
+
+
+�������
+1
+��m
+j=1
+cj
+Ij
+1
+� ∂W
+∂I1
+�������
+,
+����
+∂W
+∂I2
+���� , . . . ,
+����
+∂W
+∂In
+����
+
+
+
+
+
+
+≤
+s0 sup0<s≤s0
+�� ∂W
+∂I
+��
+∞ ≤ s0
+�� ∂W
+∂I
+��
+G,ρ,∞
+Where we have used again that on the domain Dρ(G) the inequality
+����m
+j=1
+cj
+Ij
+1
+��� ≥ 1 holds. Similarly, |I(s0) − I0| ≤ s0∥ ∂W
+∂φ ∥G,ρ,1, and hence
+|Φt − Id|G,ρ−tδ,c ≤ t∥DW∥G,ρ,c.
+Because
+(6.10)
+|φ(s) − φ0|∞ ≤ t∥ ∂W
+∂I ∥G,ρ,∞ ≤ t
+ˆδc
+c ≤ tδ1 c
+c = tδ1
+∀0 < s ≤ s0
+|I(s) − I0|1 ≤ t∥ ∂W
+∂φ ∥G,ρ,1 ≤ tˆδc ≤ tδ2
+∀0 < s ≤ s0
+hence, (φ(s), I(s)) ∈ Dρ−tδ+tδ(G) = Dρ(G) for all 0 < s ≤ s0.
+(2) Repeat the same argument as in 6.10 with φ(−s). If (φ0, I0) ∈ Dρ′−tδ,
+then (φ(−s), I(−s)) ∈ Dρ′−tδ+tδ(G) = Dρ′. Hence,
+Dρ′(G) ⊃ Φ−1(Dρ′−tδ(G)),
+then Φ(Dρ′(G)) ⊃ Dρ′−tδ(G).
+(3) Consider f an analytical function. By the previous construction f ◦ Φt
+is defined in Dρ−tδ(G). Because W is analytic we also have that f ◦ Φt
+is analytic and we can expand its Lie series. Let m ∈ Z, l ≥ m + 1, j =
+m + 1, . . . , l then
+∥Lj
+W f∥G,ρ−(j−m)tη
+≤
+2
+c∥D(Lj−1
+W f)∥G,ρ−(j−m)tη,c∥DW∥G,ρ,c
+≤
+2
+tˆηc ∥Lj−1
+W f∥G,ρ−(j−1−m)tη∥DW∥G,ρ,c,
+where we used lemma 6.20 and defined η =
+δ
+(l−m) and ˆηc = min(cη1, η2).
+Then,
+∥Ll
+Wf∥G,ρ−tδ
+≤
+�
+2∥DW∥G,ρ,c
+tˆηc
+�l−m
+≤
+el−m · (l − m)!
+�
+2∥DW∥G,ρ,c
+ˆδc
+�l−m
+∥Lm
+Wf∥G,ρ,
+where we used that ˆηc =
+ˆδc
+l−m and (l − m)(l−m) ≤ el−m · (l − m)! And
+hence, the bound for ∥rm(f, W, t)∥G,ρ−tδ is
+∞
+�
+l=m
+tl
+l!∥Ll
+W f∥G,ρ−tδ ≤
+� ∞
+�
+l=m
+(l − m)!
+l!
+�2e∥DW∥G,ρ,c
+ˆδc
+�l−m�
+· tm∥Lm
+Wf∥G,ρ
+
+6.2. TECHNICAL RESULTS
+61
+and this series converges if ∥DW∥G,ρ,c ≤
+ˆδc
+2e.
+□
+Theorem 6.25. [Iterative Lemma] H(φ, I) = ˆh(I)+R(φ, I) where ˆh(I) is as in
+equation 6.1 defined on Dρ(G). Let ˆu = ∂ˆh
+∂I and u = ∂h
+∂I , and assume u is α, K, c, ˆq-
+non-resonant. Assume that
+�� ∂
+∂I u
+��
+G,ρ2 ≤ M ′. Let δ < ρ and c > 0, A = 1 + 2Mc
+α .
+Assume that ρ2 ≤
+α
+2MK , ∥DR∥G,ρ,c ≤ αˆδc
+74A. Then, there exists a real analytic map
+Φ : Dρ− δ
+2 (G) → Dρ(G), such that H ◦ Φ = ˆh + ˜R,with:
+(1) ∥D ˜R∥G,ρ−δ,c ≤ e−Kδ1∥DR∥G,ρ,c + 14A
+αˆδc ∥DR∥2
+G,ρ,c,
+(2) |Φ − Id|G,ρ− δ
+2 ,c ≤ 2A
+α ∥DR∥G,ρ,c,
+(3) Φ(Dρ′(G)) ⊃ Dρ′− δ
+2 (G) for ρ′ ≤ ρ − δ
+2
+Proof. Recall that
+�� ∂
+∂I u
+��
+G,ρ2 ≤ M ′ implies
+�� ∂
+∂I ( ¯Bu + ¯
+A)
+��
+G,ρ2 ≤ M by equa-
+tion 6.8. By equation 6.6
+R(q) = R(q−1)
+>K
++ r2(ˆh(q−1), W (q), 1) + r1(R(q−1), W (q), 1).
+To simplify the notation we are going to omit the index of the iteration:
+(6.11)
+˜R = R>K + r2(ˆh,W, 1) + r1(R, W, 1).
+Where W is defined in terms of its Fourier expressions by equation 6.7:
+Wk(I) =
+Rk(I)
+i(k ¯B(I1)u + k ¯
+A(I1))
+By proposition 6.19: ∥DW∥G,ρ,c ≤ 2A
+α ∥DR∥G,ρ,c ≤ 2A
+α
+αˆδc
+74A =
+ˆδc
+37. And Φ is
+defined as in lemma 6.20: Φ : Dρ− δ
+2 (G) → Dρ(G).
+(1) Differentiating equation 6.11 we obtain:
+D ˜R = DR>K + Dr2(ˆh, W, 1) + Dr1(R, W, 1).
+Taking norms at every side of the expression:
+∥D ˜R∥G,ρ−δ,c
+=
+∥DR>K + Dr2(ˆh, W, 1) + Dr1(R, W, 1)∥G,ρ−δ,c
+≤
+∥DR>K∥G,ρ−δ,c + ∥Dr2(ˆh, W, 1)∥G,ρ−δ,c
++∥Dr1(R, W, 1)∥G,ρ−δ,c
+≤
+e−Kδ1∥DR∥G,ρ,c
++ 2c
+ˆδc
+�
+∥r2(ˆh, W, 1)∥G,ρ− δ
+2 ,c + ∥r1(R, W, 1)∥G,ρ− δ
+2 ,c
+�
+Let us further develop the two last terms of the previous expression,
+by using lemma 6.24:
+∥r2(ˆh, W, 1)∥G,ρ− δ
+2 ,c
+≤
+γ2
+�
+2e∥DW∥G,ρ,c
+ˆδc/2
+�
+∥L2
+W h∥G,ρ
+≤
+γ2
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+∥{{h, W}, W}∥G,ρ,
+∥r1(ˆh, W, 1)∥G,ρ− δ
+2 ,c
+≤
+γ1
+�
+2e∥DW∥G,ρ,c
+ˆδc/2
+�
+∥L1
+W R∥G,ρ
+≤
+γ1
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+∥{R, W}∥G,ρ.
+
+62
+6. A NEW KAM THEOREM
+Then, using the second statement of lemma 6.20 and that {W, h} =
+R≤K:
+∥{R, W}∥G,ρ ≤ 2
+c∥DR∥G,ρ,c∥DW∥G,ρ,c, and
+|{{h, W}, W}∥G,ρ
+=
+∥{R≤K, W}∥G,ρ
+≤
+2
+c∥DR≤K∥G,ρ,c∥DW∥G,ρ,c
+≤
+2
+c∥DR∥G,ρ,c∥DW∥G,ρ,c.
+Moreover, it is easy to see that γ1(x) =
+− log(1−x)
+x
+and γ2(x) =
+x+(1−x) log(1−x)
+x2
+. Observe that these functions are monotonously increas-
+ing in x. Recall that ∥DW∥G,ρ,c ≤ 2A
+α ∥DR∥G,ρ,c. Then,
+∥r1(ˆh, W, 1)∥G,ρ− δ
+2 ,c
++∥r2(ˆh, W, 1)∥G,ρ− δ
+2 ,c
+≤
+γ1
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+∥{R, W}∥G,ρ
++γ2
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+∥{{h, W}, W}∥G,ρ
+≤
+γ1
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+2
+c∥DR∥G,ρ,c∥DW∥G,ρ,c
++γ2
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+2
+c∥DR∥G,ρ,c∥DW∥G,ρ,c
+≤
+γ1
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+2
+c
+2A
+α ∥DR∥2
+G,ρ,c
++γ2
+�
+4e∥DW∥G,ρ,c
+ˆδc
+�
+2
+c
+2A
+α ∥DR∥2
+G,ρ,c
+≤
+2
+c[γ1( 4e
+37) + γ2( 4e
+37)] 2A
+α ∥DR∥2
+G,ρ,c
+=
+4A
+αc [γ1( 4e
+37) + γ2( 4e
+37)]∥DR∥2
+G,ρ,c.
+Moreover γ1( 4e
+37) + γ2( 4e
+37) ≈ 1.741 . . . < 7
+4.
+Then,
+∥D ˜R∥G,ρ−δ,c
+≤
+e−Kδ1∥DR∥G,ρ,c + 2c
+ˆδc
+4A
+αc
+7
+4∥∥2
+G,ρ,c
+≤
+e−Kδ1∥DR∥G,ρ,c + 14A
+ˆδcα ∥DR∥2
+G,ρ,c,
+as we wanted to prove.
+(2) Direct from lemma 6.24:
+|Φ − Id|G,ρ. δ
+2 ,c ≤ ∥DW∥G,ρ,c ≤ 2A
+α ∥DR∥G,ρ,c
+(3) Also direct from lemma 6.24:
+Φ(Dρ(G)) ⊃ Dρ′− δ
+2 (G), for ρ′ ≤ ρ − δ/2
+□
+Definition 6.26. ∆c,ˆq(k, α) = {J ∈ Rn such that |k ¯B(I1)J + k ¯
+A(I1)| < α}
+Lemma 6.27. With the previous definitions we have the following bounds.
+Outside of Z:
+meas (F ∩ ∆c,ˆq(k, α)) ≤ (diamF)n−1 2α
+|k|2,ω
+.
+
+6.2. TECHNICAL RESULTS
+63
+At Z:
+meas (F ∩ ∆c,ˆq(k, α))
+�
+= 0
+if α ≤ |k1|
+K′
+≤ (diamF)n
+if α > |k1|
+K′
+Proof. It is important to understand the geometry of the set ∆c,ˆq(k, α). Re-
+call that k ¯B(I1)J = k1B(I1)J1 + ¯k ¯J, hence this part of the expression can be inter-
+preted as the scalar product of the vector J with the vector (k1B(I1), k2, . . . , kn).
+Then the set {J ∈ Rn such that |k ¯B(I1)J| < α} is the space between two hyper-
+planes orthogonal to (k1B(I1), k2, . . . , kn). Adding the term k ¯
+A(I1) only applies
+a transition to the previous set. Let us find what is the separation between the
+hyperplanes. Assume J is parallel to (k1B(I1), k2, . . . , kn) with lengths a:
+J = a(k1B(I1), k2, . . . , kn)
+|k|2,ω
+,
+where |k|2,ω =
+�
+B(I1)2k2
+1 + k2
+2 + . . . k2n. Then,
+J · (B(I1), k1, . . . , kn)
+=
+c(B(I1)k2
+1 + k2
+2 + . . . k2
+n)
+1
+|k|2,ω
+=
+a|k|2,ω ≤ α ⇔ a ≤
+α
+|k|2,ω .
+And finally,
+meas (F ∩ ∆c,ˆq(k, α)) ≤ (diamF)n−1 2α
+|k|2,ω
+.
+The previous formula can not be applied if when we are at Z and k = (k1, 0, . . . , 0).
+(B(I1)k1, k2, . . . , kn)
+|k ¯B(I1)J + k ¯
+A(I1)| < α
+|k ¯B(I1)J| < α
+−k ¯
+A(I1)
+Figure 1. Graphical representation of the set ∆c,ˆq(α)
+At Z,
+∆c,ˆq(K, α) = {J ∈ Rn such that | ¯K ¯J + k1
+ˆqm
+cm
+| < α}.
+And if k = (k1, 0, . . . , 0) then
+∆c,ˆq(K, α) = {J ∈ Rn such that |k1
+ˆqm
+cm
+| < α}.
+
+64
+6. A NEW KAM THEOREM
+Then
+∆c,ˆq(k, α) =
+� Rn
+if |k1| < α cm
+ˆqm = αK′,
+{∅}
+if |k1| ≥ α cm
+ˆqm = αK′.
+Using this last identity, the statement we wanted to prove is immediate.
+□
+Definition 6.28. G − b := {I ∈ G such that Ub(I) ⊂ G}, where Ub(I) is the
+ball of radius b centered at I.
+Definition 6.29. F is a D-set if meas[(F − b1) \ (F − b2)] ≤ D(b2 − b1).
+Lemma 6.30. Let F ⊂ Rn be a D-set for d ≥ 0, τ > 0, β ≥ 0 and k ≥ 0 an
+integer. Consider the set
+F(d, β, K) := (F − d) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|τ
+1
+�
+.
+Then, outside of Z:
+(1) If d′ ≥ d, β′ ≥ β, k′ ≥ k, then
+meas[F(d, β, k) \ F(d′, β′, k′)] ≤
+D(d′ − d) + 2(diamF)n−1
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+β′ − β
+|k|τ
+1|k|2,ω
++
+�
+k∈Zn\{0}
+0<|k|1≤K
+β′
+|k|τ
+1|k|2,ω
+
+
+
+
+(2) For every b ≥ 0
+meas[F(d, β, K) \ (F(d, β, K) − b)] ≤ (D + 2n+1(dim F)n−1Kn)b
+And inside of Z, if we assume β ≤
+1
+K′ , the equation 1 holds adding only the terms
+¯k ̸= 0 and 2 holds without any change.
+Proof. Recall that
+∆c,ˆq
+�
+k, β
+|k|τ
+1
+�
+=
+�
+J ∈ Rn such that
+��k ¯B(I1)J + k ¯
+A(I1)
+�� <
+β
+|k|τ
+1
+�
+.
+First we will prove the results outside of Z and then
+(1) Let us expand the expression of meas[F(d, β, k) \ F(d′, β′, k′)]:
+
+(F − d) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|τ
+1
+�
+
+ \
+
+(F − d) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|τ
+1
+�
+
+ .
+Now we use the following property on the previous expression:
+(A \ B) \ (C \ D)
+=
+[(A \ B) \ C] ∪ [(A \ B) ∩ D]
+⊂
+(A \ C) ∪ [(A \ B) ∩ D]
+=
+(A \ C) ∪ (A ∩ (D \ B)),
+where the last equality holds true because D ⊃ B. Using this property
+we have that meas[F(d, β, k) \ F(d′, β′, k′)] is included in
+
+6.2. TECHNICAL RESULTS
+65
+[(F − d) \ (F − d′)]
+∪
+
+(F − d) ∩
+
+
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K′
+∆c,ˆq
+�
+k, β′
+|k|1
+�
+
+
+
+
+\
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|1
+�
+
+
+
+
+
+
+
+ .
+And this expression is equivalent to:
+[(F − d) \ (F − d′)]
+∪
+�
+k∈Zn\{0}
+|k|1≤K
+�
+(F − d) ∩
+�
+∆c,ˆq
+�
+k, β′
+|k|τ
+1
+�
+\∆c,ˆq
+�
+k, β
+|k|τ
+1
+���
+∪
+�
+k∈Zn\{0}
+K<|k|1≤K′
+�
+(F − d) ∩ ∆c,ˆq
+�
+k, β′
+|k|τ
+1
+��
+.
+Now, using lemma 6.27 we obtain:
+meas(F(d, β, K) \ F(d′, β′, K′)) ≤
+≤ D(d′ − d) + (diamF)n−1
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+2(β′ − β)
+|k|τ
+1|k|2,ω
++
+�
+k∈Zn\{0}
+K<|k|1≤K′
+2β′
+|k|τ
+1|k|2,ω
+
+
+
+
+(2) Observe that:
+F(d, β, K) − b
+=
+
+(F − d) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|τ
+1
+�
+
+ − b
+⊃
+(F − (d + b)) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k, β
+|k|τ
+1
++ b|k|2,ω
+�
+.
+Then,
+meas[(F(d, β, K)) \ (F(d, β, K) − b)]
+≤
+meas
+��
+(F − d) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k,
+β
+|k|τ
+1
+��
+\
+�
+(F − (d + b)) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq
+�
+k,
+β
+|k|τ
+1
+���
+≤
+meas [(F − d) \ (F − (d + b))∪
+
+66
+6. A NEW KAM THEOREM
+�
+k∈Zn\{0}
+|k|1≤K
+�
+(F − d) ∩
+�
+∆c,ˆq
+�
+k,
+β
+|k|τ
+1 + b|k|2,ω
+���
+\
+�
+∆c,ˆq
+�
+k,
+β
+|k|τ
+1
+���
+≤
+Db + �
+k∈Zn\{0}
+|k|1≤K
+(diamF)n−1 2b|k|2,ω
+|k|2,ω
+≤
+Db + 2nKn(diamF)n−1 · 2 = Db + 2n+1Kn(diamF)n−1,
+where in the last inequality we used that the number of vectors k such
+that |k|1 ≤ K is less or equal than 2nKn.
+The previous identities worked outside of Z. Let us understand the set F(d, β, K)
+when we are ate Z.
+F(d, β, K)
+:=
+(F − d) \ �
+k∈Zn\{0}
+|k|1≤K
+∆cˆq(k,
+β
+|k|τ
+1 )
+=
+(F − d) \
+
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+¯k̸=0
+∆cˆq(k,
+β
+|k|τ
+1 )
+
+
+
+∪
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+¯k=0
+∆cˆq(k,
+β
+|k|τ
+1 )
+
+
+
+
+
+=
+(F − d) \
+
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤K
+¯k̸=0
+∆cˆq(k,
+β
+|k|τ
+1 )
+
+
+
+∪
+
+�
+k1∈Z\{0}
+|k|1≤
+β
+|k1|τ K′
+Rn
+
+
+
+.
+Note that if for some k1 ∈ Z \ {0}, |k|1 ≥
+β
+|k|τ
+1 K′, we take out all the possible
+frequencies. Then seems natural to ask |k|1 ≥
+β
+|k|τ
+1 K′ for all k1 ∈ Z \ {0}, which
+holds if and only if |k1|1+τ ≥ βK′ for all k1 ∈ Z \ {0} or simply β ≤
+1
+K′ which we
+assumed. Then
+F(d, β, K) := (F − d) \
+�
+k∈Zn\{0}
+|k|1≤K
+¯k̸=0
+∆cˆq(k, β
+|k|τ
+1
+).
+Hence we can replicate the proof of 1 only with the terms ¯k ̸= 0. And the bound
+of 2 can be slightly improved by using that the number of vectors k ∈ Zn \ {0}
+such that |k|1 ≤ K and |¯k| ̸= 0 is bounded by 2nKn − K, but since it is not a big
+improve, for the sake of simplicity we assume the bound 2 at Z.
+□
+Lemma 6.31. Let G ⊂ Rn be compact.
+u, ˜u : G → Rn maps of class C2.
+|˜u − u| ≤ ε.
+Assume that u is one-to-one on G, let F = u(G).
+Consider the
+following bounds:
+����
+∂u
+∂I
+����
+G
+≤ M,
+����
+∂u
+∂I (I) · v
+���� ≥ µ|v|
+∀v ∈ Rn, ∀I ∈ G,
+
+6.2. TECHNICAL RESULTS
+67
+����
+∂˜u
+∂I
+����
+G
+≤ ˜
+M,
+����
+∂˜u
+∂I2
+����
+G
+≤ ˜
+M2,
+����
+∂˜u
+∂I (I)v
+���� ≥ ˜µ|v|
+∀v ∈ Rn, ∀I ∈ G,
+˜µ < µ and ˜
+M < M. Assume ε ≤ ˜mu2/(4 ˜
+M2). Then, given a subset ˜F ⊂ F − 4Mε
+˜µ
+and writing ˜G = (˜u)−1( ˜F ), the map ˜u is one-to-one from ˜G to ˜F and
+˜G ⊂ G − 2ǫ
+˜µ ,
+u( ˜G) ⊃ ˜F − ε.
+Moreover,
+|(˜u)−1 − u−1| ˜
+F ≤ ε
+µ
+Proof. The statement is not any different than the classical one, so we are
+not going to prove it in here. A proof can be found in [DG96].
+□
+Lemma 6.32 (Inductive lemma). Let G ⊂ Rn be a compact.
+H(φ, I) = ˆh(I) + R(φ, I)
+where ˆh is defined as in 6.1 in the domain Dρ(G),and R(φ, I) analytic on the same
+domain. Let ˆu = ∂ˆh
+∂I and u = ∂h
+∂I . Assume that | ∂
+∂I u|G,ρ2 ≤ M ′ and |u|G ≤ L. Also,
+assume that u is non-degenerate:
+����
+∂u
+∂I v
+���� ≥ µ|v|
+∀I ∈ G.
+Let ˜
+M > M, ˜L > L and ˜µ < µ. Assume u is one-to-one on G and denote F = u(G).
+Assume τ > 0, 0 < β ≤ 1 and K given. Assume also that
+F ∩ ∆c,ˆq
+�
+K, β
+|k|τ
+1
+�
+= ∅,
+∀k ∈ Zn, |k|1 ≤ K, k ̸= 0.
+Denote ǫ := ∥DR∥G,ρ,c, η := |R0|G,ρ2 and ξ :=
+�� ∂R0
+∂I
+��
+G,ρ2.
+(1) ρ2 ≤
+β
+2MKτ+1
+(2) ǫ ≤ min
+�
+βˆδc
+74AKτ , ˜µ2(ρ2−δ2)
+4 ˜
+M
+�
+(3) ξ ≤ min
+�
+( ˜
+M − M)δ2/R, (µ − ˜µ)ρ2
+�
+Then there exists a real canonical transformation
+Φ : Dρ− δ
+2 (G) → Dρ(G)
+and a decomposition H ◦ Φ = ˜ˆh(I) + ˜R(φ, I). Writing ˜u =
+∂
+∂I ˜h one has.
+(1) |˜u − u|G,ρ2 = ξ,
+|˜h − h|G,ρ2 = η,
+(2) ˜ǫ := ∥D ˜R∥G,ρ−δ,c ≤ e−Kδ1ǫ + 14AKτ
+βˆδc
+ǫ2,
+(3) ˜η := | ˜R0|G,ρ2− δ2
+2 ≤ 7AKτ
+cβ
+ǫ2,
+(4) |Φ − Id|G,ρ− δ
+2 ,c ≤ 2AKτ
+β
+ǫ,
+(5)
+�� ∂
+∂I ˜u
+��
+G,ρ2 ≤ ˜
+M ′, |˜u|G ≤ ˜L,
+(6) | ∂˜u
+∂I v| ≥ ˜µ|v|
+∀I ∈ G,
+(7) Given a subset ˜F ⊂ F − 4Mǫ
+˜µ , ˜G(˜u)−1( ˜F) the map ˜u is one-to-one from
+˜G to ˜F, ˜G ⊂ G − 2ǫ
+˜µ , u( ˜G) ⊃ ˜F − ǫ. Moreover |˜u−1 − u−1| ˜
+F ≤ ǫ/µ.
+
+68
+6. A NEW KAM THEOREM
+Proof. The set u(I) is β/Kτ, K-non-resonant with respect to ω. This implies
+that
+(6.12)
+|k1B(I1)u1 + ¯k¯u + A(I1)u1| ≥ β/Kτ. ≥
+β
+|k|τ
+1
+≥ β
+Kτ .
+Then ρ2 ≤ β/Kτ
+2MK =
+β
+2MKτ+1 , ∥DR∥G,ρ,c ≤ β/Kτ ˆδc
+74A
+=
+βˆδc
+74AKτ . We apply the iterative
+lemma (Theorem 6.25) to obtain Φ : Dρ− δ
+2 (G) → Dρ(G), such that H ◦ Φ = ˜h + ˜R
+where ˜h = h + R0.
+We have taken out the points that are not β/Kτ, K-non-resonant with respect
+to ω. Because of conditions 1 and 2 we can apply the Iterative lemma. Now let us
+prove each of the points in the statement.
+(1) We know by definition that ˜u = ∂˜h
+∂I = ∂(h+R0)
+∂I
+= ∂h
+∂I + R0
+∂I , hence:
+|˜u − u|G,ρ2 = |∂h
+∂I + ∂R0
+∂I − ∂h
+∂I |G,ρ2 = |∂R0
+∂I |G,ρ2 = ξ
+˜h = h + R0 ⇒ |˜h − h|G,ρ2 = |h + R0 − h|G,ρ2 = |R0|G,ρ2 = η
+(2) By the iterative lemma:
+∥D ˜R∥G,ρ−δ,c
+≤
+e−Kδ1∥DR∥G,ρ,c + 14A
+αˆδc ∥DR∥G,ρ,c
+≤
+e−Kδ1ε + 14A
+αˆδc ε2
+=
+e−Kδ1ε + 14AKτ
+βˆδc
+ε2,
+where we have used that α =
+β
+Kτ .
+(3) At this point we use an inequality used in the proof of the iterative Lemma
+(theorem 6.25).
+| ˜R0|G,ρ2−δ2/2
+≤
+|r2(h, W, 1) + r1(R, W, 1)|G,ρ2−δ2/2
+≤
+7A
+αc ∥DR∥2
+G,ρ,c = 7AKτ
+β
+ε2.
+(4) Also using the the iterative Lemma:
+|Φ − id|G,ρ−δ/2,c ≤ 2A
+α ∥DR∥G,ρ,c = 2AKτ
+β
+∥DR∥G,ρ,c.
+(5) Recall that | ∂
+∂I Aω˜u|G,ρ2−δ2 ≤ ˜
+M, |˜u|G ≤ ˜L, ˜h = h + R0, | ∂
+∂I Aωu|G,ρ2 ≤
+M, |u|G ≤ L. Note that A(I1) ≤ m · maxj(qj)/ minj(cj) and B(I1) ≤
+1/ minj(cj). Hence A(I1) + B(I1) ≤ maxj(qj)/ minj(cj) + 1/ minj(cj) :=
+R, and we have that |Aω| ≤ R.
+| ∂
+∂I Aω˜u|G,ρ2−δ2
+=
+| ∂
+∂I Aω˜u + ∂
+∂I Aωu − ∂
+∂I Aωu|G,ρ2−δ2
+≤
+| ∂
+∂I Aω(˜u − u)|G,ρ2−δ2 + | ∂
+∂I Aωu|G,ρ2−δ2
+≤
+| ∂
+∂I AωR0|G,ρ2−δ2 + M
+≤
+|Aω|G,ρ2|R0|G,ρ
+δ2
++ M
+≤
+|Aω|G,ρ2·ξ
+δ2
++ M
+≤
+Rξ
+δ2 + M
+≤
+( ( ˜
+M−M)δ2
+R
+)R
+δ2
++ M ≤ ˜
+M − M + M = ˜
+M,
+where ξ ≤ ( ˜
+M − M)δ2/R.
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+69
+(6) We know | ∂u
+∂I (I)v| ≥ µ|v| for all I ∈ G, then | ∂u
+∂I (I)v|G ≥ µ|v|. We want
+to find | ∂˜u
+∂I (I)v|G ≥ µ′|v| if µ′ < µ.
+| ∂˜u
+∂I v|G
+=
+|( ∂˜u
+∂I + ∂u
+∂I − ∂u
+∂I )v|G
+=
+|( ∂2R0
+∂I2 + ∂u
+∂I )v|G
+≥
+−| ∂2R0
+∂I2 v|G + | ∂u
+∂I v|G
+≥
+µ|v| − | ∂2R0
+∂I2 |G|v|
+≥
+µ|v| − | ∂R0
+∂I |G 1
+δ2 |v|
+≥
+µ|v| − ξ
+ρ2 |v| = (µ − ξ/ρ2)|v| ≥ µ′|v|,
+where we have used that | ∂2R0
+∂I2 |G ≤ | ∂R0
+∂I | 1
+ρ2 , and also that µ′ < µ − ξ/ρ2,
+hence ξ ≤ (µ − µ′)ρ2.
+(7) To apply lemma 6.31 we only need to check that ε ≤
+˜µ2
+˜
+M2 .
+˜
+M2 can be
+chosen such that | ∂2u
+∂I2 |G ≤ ˜
+M2. Note that | ∂2u
+∂I2 |G ≤ | ∂2u
+∂I2 |G,ρ2−δ2.
+| ∂u
+∂I |G,ρ2−δ2 ≤ ˜
+M
+⇒
+| ∂2u
+∂I2 |G,ρ2−δ2(ρ2 − δ2) ≤ | ∂u
+∂I |G,ρ2−δ2 ≤ ˜
+M
+⇒
+| ∂2u
+∂I2 |G,ρ2−δ2 ≤
+˜
+M
+ρ2−δ2 = ˜
+M2
+⇒
+| ∂2u
+∂I2 |G ≤ ˜
+M2
+Then ε ≤
+˜
+M
+4 ˜
+M2 if and only if ε ≤ µ2/(4
+˜
+M
+(ρ2−δ2)) if and only if ε ≤ µ2(ρ2−δ2)
+4 ˜
+M
+which it is assumed in the statement.
+□
+6.3.
+A KAM theorem on bm-symplectic manifolds
+Theorem B ( A bm-KAM theorem). Let G ⊂ Rn, n ≥ 2 be a compact set. Let
+H(φ, I) = ˆh(I)+f(φ, I), where ˆh is a bm-function ˆh(I) = h(I)+q0 log(I1)+�m−1
+i=1
+qi
+Ii
+1
+defined on Dρ(G), with h(I) and f(φ, I) analytic. Let ˆu = ∂ˆh
+∂I and u = ∂h
+∂I . Assume
+| ∂u
+∂I |G,ρ2 ≤ M, |u|G ≤ L. Assume that u is µ non-degenerate (| ∂u
+∂I v| ≥ µ|v| for some
+µ ∈ R+ and I ∈ G. Take a = 16M. Assume that u is one-to-one on G and its range
+F = u(G) is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1. Let
+(1)
+(6.13)
+ε := ∥f∥G,ρ ≤
+ν2µ2ˆρ2τ+2
+24τ+32L6M 3 γ2,
+(2)
+(6.14)
+γ ≤ min(8LMρ2
+ν ˆρτ+1 , L
+K′ )
+(3)
+(6.15)
+µ ≤ min(2τ+5L2M, 27ρ1L4Kτ+1, βντ+122τ+1ρτ
+1),
+where ˆρ := min
+�
+νρ1
+12(τ+2), 1
+�
+. Define the set ˆG = ˆGγ := {I ∈ G− 2γ
+µ |u(I) is τ, γ, c, ˆq−
+Dioph.}. Then, there exists a real continuous map T : W ρ1
+4 (Tn) × ˆG → Dρ(G) an-
+alytic with respect the angular variables such that
+(1) For all I ∈ ˆG the set T (Tn ×{I}) is an invariant torus of H, its frequency
+vector is equal to u(I).
+
+70
+6. A NEW KAM THEOREM
+(2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)) with estimates
+|Tφ(φ, I)| ≤ 22τ+15ML2
+ν2ˆρ2τ+1
+ε
+γ2
+|TI(φ, I))| ≤ 210+τL(1 + M)
+ν ˆρτ+1
+ε
+γ
+(3) meas[(Tn ×G)\T (Tn× ˆG)] ≤ Cγ where C is a really complicated constant
+depending on n, µ, D, diamF, M, τ, ρ1, ρ2, K and L.
+Proof. This proof, as the one in [DG96] is going to be structured in six
+sections. First we define the parameters used in each iteration while building the
+diffeomorphism. After that, we prove that we can apply the inductive lemma 6.32
+and we exhibit some bound that hold using the results of the inductive lemma. Next,
+we find that the sequence of frequency vectors and the sequence of diffeomorphisms
+that we built actually converges. Then we find estimates of the components of the
+canonical transformation that we have built. Then we find a way to identify the
+invariant tori and finally we give a bound for the measure of the set of invariant
+tori.
+(1) Choice of parameters
+We are going to make iterative use of proposition 6.32. So we need
+to properly define all the parameters in the statement for every iteration.
+Let:
+
+
+
+Mq
+=
+(2 − 1
+2q )M,
+Lq
+=
+(2 − 1
+2q )L,
+µq
+=
+(1 + 1
+2q ) µ
+2 .
+Note that Mq, Lq monotonically increase from M to 2M and L to 2L
+when q → ∞. On the other hand µq monotonically decreases from µ to
+µ/2. Also, let:
+� K0
+=
+0,
+Kq
+=
+K · qq−1, q ≥ 1,
+where K is the minimum natural number greater or equal than 1/ˆρ
+and greater or equal than (
+νβ
+µ22τ+12 )1/τ. Moreover, β := γ/L ≤ 1, and
+
+
+
+
+
+ρ(q)
+=
+(ρ(q)
+1 , ρ(q)
+2 ),
+ρ(q)
+1
+=
+(1 +
+1
+2νq ) ρ1
+4 ,
+ρ(q)
+2
+=
+νβ
+32MKτ+1
+q+1 .
+Notice that ρ(q)
+1
+decreases monotonically from ρ1/2 to ρ1/4. Also, ρ(q)
+2
+decreases to 0. We also denote:
+
+
+
+
+
+
+
+δ(q)
+1
+=
+ρ(q−1)
+1
+− ρ(q)
+1 ,
+δ(q)
+2
+=
+ρ(q−1)
+2
+− ρ(q)
+2 ,
+cq
+=
+δ(q)
+2
+δ(q)
+1 .
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+71
+Note that
+δ(q)
+1
+=
+�
+1 +
+1
+2ν(q−1
+� ρ1
+4 −
+�
+1 +
+1
+2νq
+� ρ1
+4
+=
+�
+1
+2ν(q−1) −
+1
+2νq
+� ρ1
+4
+=
+1−1/2ν
+2ν(q−1)
+ρ1
+4 .
+Also, since 0 < ν < 1 then ν/2 ≤ 1 − 1/2ν ≤ ν. Plugging this in the
+previous equation we obtain:
+(6.16)
+νρ1
+2ν(q−1)8 ≤ δ(q)
+1
+≤
+νρ1
+2ν(q−1)4.
+Also,
+δ(q)
+2
+=
+νβ
+32MKτ+1
+q
+−
+νβ
+32MKτ+1
+q+1
+=
+νβ
+32M(K2q−1)τ+1 −
+νβ
+32M(K2q)τ+1
+=
+νβ
+32M(K2q−1)τ+1
+�
+1 −
+1
+2τ+1
+�
+.
+Also, since τ > 0 then 1/2 ≤ (1 − 1/2τ+1) ≤ 1. Using this in the
+previous equation:
+(6.17)
+νβ
+64MKτ+1
+q
+≤ δ(q)
+2
+≤
+νβ
+32MKτ+1
+q
+.
+Using equations 6.16 and 6.17 we find bounds for cq
+
+
+
+
+
+
+
+
+
+
+
+cq
+≤
+�
+νβ
+32MKτ+1
+q
+�
+�
+νρ1
+2ν(q−1)
+�
+=
+β2ν(q−1)
+4MKτ+1
+q
+ρ1 ,
+cq
+≥
+�
+νβ
+64MKτ+1
+q
+�
+�
+νρ1
+2ν(q−1)4
+� =
+β2ν(q−1)
+16MKτ+1
+q
+ρ1 .
+Then, we also define
+� βq
+=
+(1 −
+1
+2νq )β,
+β′
+q
+=
+βq+βq+1
+2
+.
+Observe that both βq and β′
+q tend to β. Also observe that β′
+q ≥ ν
+4β,
+because:
+β′
+q
+=
+βq+βq+1
+2
+=
+(1−
+1
+2νq )+�
+1−
+1
+2ν(q+1)
+�
+2
+β
+=
+�
+1 −
+� 1+ 1
+2ν
+2νq
+�
+1
+2
+�
+β
+≥
+�
+1 − (1 − 1/2ν) 1
+2
+�
+β ≥ ν
+4β.
+As K is the minimal natural number such that K ≥ 1/ˆρ then K ≤
+2/ˆρ. Hence ˆρ ≤ 2
+K . Also
+1
+ˆρτ+1 ≥
+�K
+2
+�τ+1
+.
+Recall that ˆρ = min(
+νρ1
+12(τ+2), 1) and, in particular, ˆρ ≤ νρ1 and ˆρ ≤ 1.
+
+72
+6. A NEW KAM THEOREM
+By definition γ ≤ 8LMρ2
+ν ˆρτ+1 . And because β = γ/L:
+βL ≤ 8LMρ2
+ν ˆρτ+1 ≤ 8LMρ2Kτ+1
+ν
+.
+Because we assumed ε ≤
+ν2µ2 ˆρ2τ+2
+24τ+32L6M3 γ2 then, using that γ = Lβ and
+ˆρ ≤ 2/K:
+(6.18)
+ε ≤ ν2µ2 � 2
+K
+�2τ+2
+24τ+32L6M 3 ≤
+ν2µ2β2
+24τ+30L4M 3K2τ+2 .
+Also using again the assumption that ε ≤
+ν2µ2 ˆρ2τ+2
+24τ+32L6M3 γ2 we want to
+prove that
+(6.19)
+ε ≤
+ν3ρ1β2
+22τ+22MK2τ+1 .
+It is enough to check that:
+ν2µ2ˆρ2τ+2L2β2
+24τ+32L6M 3
+≤
+ν3ρ1β2
+22τ+22MK2τ+1
+where we used γ = Lβ. Now observing that ˆρ ≤ νρ1 it suffices to see
+ν2µ2ν2τ+2ρ2τ+2
+1
+L2β2
+24τ+32L6M 3
+≤
+ν3ρ1β2
+22τ+22MK2τ+1 ,
+which simplifies to
+µ2ρ2τ+2
+1
+24τ+10L4M 2 ≤
+1
+K2τ+1 .
+Using that K ≥ 1/(νρ1) is enough to check that
+µ2ρ2τ+1
+1
+ν2τ+2
+22τ+12L4M 2 ≤ (νρ1)2τ+1,
+which holds if and only if µ ≤ 2τ+5L2M as we assumed.
+(2) Induction
+Let us take G0 = G. Now the goal is to construct a decreasing se-
+quence of compact sets Gq ⊂ G and a sequence of real analytic canonical
+transformations
+Φ(q) : Dρ(q)(Gq) → Dρ(q−1)(Gq−1),
+q ≥ 1.
+Denoting Ψ(q) = Φ1 ◦· · ·◦Φ(q) the transformed Hamiltonian functions
+will be noted by H(q) = H ◦ Ψ(q) = ˆh(q)(I) + R(q)(φ, I). Moreover, u(q) =
+∂h(q)
+∂I
+and ˆu(q) = ∂ˆh(q)
+∂I .
+We are going to show that the following bounds hold for all q ≥ 0:
+(a) εq := ∥DR(q)∥Gq,ρ(q),cq+1 ≤
+8ε
+νρ12(2τ+2)q ,
+(b) ηq := |R(q)
+0 |Gq,ρ(q)
+2
+≤
+ε
+2(2τ+3)q and ξq := | ∂R(q)
+0
+∂I |Gq,ρ(q)
+2
+≤ 4MKτ+1ε
+νβ2(τ+2)q ,
+(c) | ∂2h(q)
+∂I2 |Gq,ρ(q)
+2
+≤ Mq,
+|u(q)| ≤ Lq
+∀I ∈ Gq,
+(d) u(q) is µq-non-degenerate on Gq,
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+73
+(e) u(q) is one-to-one on Gq, and u(q)(Gq) = Fq where we define:
+Fq := (F − βq) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆cq,ˆq(K, βq
+|k|τ
+1
+)
+To prove this we proceed by induction. For q = 0:
+
+
+
+G0 = G,
+h(0) = h, ˆh(0) = ˆh,
+R(0) = f.
+Using the definitions from the previous point:
+�
+ρ(0)
+1
+=
+(1 + 1) ρ1
+4 = ρ1/2,
+ρ(0)
+2
+=
+νβ
+32MKτ+1 ≤ ρ2
+2 ,
+where in the last inequality we have used that β ≤
+8Mρ2Kτ+1
+ν
+and
+hence ρ2 ≥
+βν
+8MKτ+1 .
+Then,
+ε0 = ∥Df∥G,ρ(0),c1 = ∥Df∥G,ρ(1)+δ(1).
+Now, let us use that |DΥ|G,ρ−δ,c ≤ 2|Υ|G,ρ,c
+ˆδc
+while having in mind that
+ˆδ(1)
+c1 = min(c1δ(1)
+1 , δ(1)
+2 ). Then,
+∥Df∥G,ρ(1),c1 ≤ c1|f|G,ρ(0)
+ˆδc1
+≤ |f|G,ρ(0)
+δ(1)
+1
+≤ |f|G,ρ(0)8
+νρ1
+= 8ε
+νρ1
+,
+where we have used δ(1)
+1
+≥
+νρ1
+8·2ν(1−1) = νρ1
+8 .
+This proves the first step of
+the induction for 2a).
+Let us prove now the base case for 2b). On one side η0 = |R(0)
+0 |G0,ρ2(0) ≤
+ε
+2(2τ+3)0 = ε, which holds because |R(0)
+0 |G0,ρ(0)
+2
+≤ |R(0)|G,ρ(0) = |f|G,ρ(0) =
+ǫ. On the other hand ξ0 = | ∂R(0)
+0
+∂I |G0,ρ(0)
+2
+≤ | ∂R(0)
+0
+∂I |G0,ρ2−ρ2/2 ≤
+1
+ρ2/2∥R0∥G,ρ ≤
+2ε
+ρ2 ≤
+ε
+ρ(0)
+2 , where we used that ρ2(0) ≤ ρ2/2 = ρ2 − ρ2/2.
+The base case of 2c) is immediate because | ∂2h(0)
+∂I2 |G0,ρ(0)
+2
+≤ | ∂2h
+∂I2 |G,ρ2 =
+M = M0 and also |u(0)|G0 = |u|G ≤ L = L0.
+The base case of 2d holds because u(0) = u is µ non-degenerate in
+G = G0.
+The base case of 2e holds because u(0) = u is one-to-one in G0 = G
+by hypothesis. u(0)(G0) = F0 where F0 = (F − β0) \ {∅} = F because
+K0 = 0 and β0 = 0.
+For q ≥ 1, we assume the statements hold for q − 1 and we prove
+it for q. Let us apply proposition 6.32 (Inductive Lemma) to H(q−1) =
+hq−1 + Rq−1 with Kq instead of K.
+
+74
+6. A NEW KAM THEOREM
+We have to be careful with the condition F ∩ ∆c,ˆq(k,
+β
+|k|τ
+1 ) = ∅ ∀k ∈
+Zn, |k|1 ≤ Kq, k ̸= 0 and with the definition
+Fq−1 := (F − βq−1) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k, βq−1
+|k|τ
+1
+),
+because the resonances have to be removed up to order Kq, not Kq−1.
+Let us define
+F ′
+q−1 := (F − βq−1) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k, β′
+q−1
+|k|τ
+1
+)
+where we simply replaced βq−1 for β′
+q−1 because ∆c,ˆq(k, βq−1
+|k|τ
+1 ) makes
+no sense when q = 1, βq−1 = 0.
+Accordingly let us define G′
+q−1 := (u(q−1))−1(F ′
+q−1). The conditions in
+proposition 6.32 are going to be satisfied with F ′
+q−1, β′
+q−1, Kq, Mq−1, Lq−1,
+µq−1,ρ(q−1), δ(q), cq, Mq, Lq, µq replacing F, β, K, M, L, µ, ρ, δ, c, ˜
+M, ˜L, ˜µ.
+And also a = 16M ≥ 8Mq.
+We are now going to check that 1, 2 and 3 are satisfied so we can
+apply proposition 6.32.
+– 1 We want to see that ρ(q−1)
+2
+≤
+β′
+q−1
+2MqKτ+1
+q
+.
+By definition ρq−1
+2
+=
+νβ
+32MKτ+1
+q
+≤
+4β′
+q−1
+32MKτ+1
+q
+≤
+β′
+q−1
+8MqKτ+1
+q
+≤
+βq−1
+2MqKτ+1
+q
+, where we used that
+Mq ≥ M.
+– 2 We want to see that εq−1 ≤ min
+�
+βq−1 ˆρ(q)
+c
+74AqKτ
+q−1 ,
+µτ
+q (ρ(q−1)
+2
+−δ(q−1)
+c
+)
+4Mq
+�
+,
+where Aq := 1 +
+2Mq−1cqKτ
+q
+β′
+q−1
+.
+Notice that:
+Aq
+:=
+1 +
+2Mq−1cqKτ
+q
+β′
+q−1
+≤
+1 +
+8Mq−1cqKτ
+q
+νβ
+≤
+1 +
+8Mq−1β2ν(q−1)Kτ
+q
+4MKτ+1
+q
+ρ1νβ
+=
+1 + 2Mq−12ν(q−1)
+MKqρ1ν
+=
+1 + 2Mq−12ν(q−1)
+MK2q−1ρ1ν
+≤
+1 +
+4M2q−1
+MK2q−1ρ1ν
+=
+1 +
+4
+Kρ1ν ≤ 1 + 4 = 5
+First, we check that εq−1 ≤
+βq−1 ˆρ(q)
+c
+74AqKτ
+q−1 .
+By induction hypothesis we know that εq−1 ≤
+8ε
+νρ12(2τ+2)(q−1) . Hence
+it is enough to see
+8ε
+νρ12(2τ+2)(q−1) ≤ β′
+q−1δ(q)
+2
+75 · 5Kτq
+.
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+75
+Notice that
+β′
+q−1δ(q)
+2
+379Kτq
+≥
+νβ
+4
+νβ
+64MKτ+1
+q
+1
+37Kτq
+=
+ν2β2
+4·64·370
+1
+MK2τ+1
+q
+=
+ν2β2
+4·64·370·M2(q−1)(2τ+1)K2τ+1 .
+And this holds if the following is true:
+8ε
+νρ1
+≤
+ν2β2
+4 · 64 · 379 · K2τ+1M ⇔ ε ≤
+ν3β2ρ1
+212135MK2τ+1.
+This is true because in the previous section we have seen that ε ≤
+ν3ρ1β2
+22τ+30MK2τ+1
+Let us now prove that ε ≤
+µ2
+q(ρ(q−1)
+2
+−δ(q−1)
+2
+)
+2Mq
+. First of all observe that
+(ρ(q−1)
+2
+− δ(q)
+2 ) = ρ(q)
+2 . So, what we want to prove is equivalent to
+proving εq−1 ≤
+µ2
+qρ(q)
+2
+2Mq .
+On the other hand, we know that εq−1 ≤
+8ε
+νρ12(2τ+2)(q−1) . And observe
+also that
+µ2
+qρ(q)
+2
+2Mq
+≥
+(µ/2)2
+νβ
+32MKτ+1
+2M
+=
+µ2νβ
+28M2Kτ+1 .
+If are able to check that
+8ε
+νρ12(2τ+2)(q−1) ≤
+µνβ
+28M2Kτ+1 we would be fine.
+The previous equation holds if and only if the following holds,
+ε ≤ µν2βρ12(2τ+2)(q−1)
+211M 2Kτ+1
+.
+If we knew beforehand that ε ≤ µν2βρ12−(2τ+2)
+211M2Kτ+1
+=
+µν2βρ1
+22τ+13M2Kτ+1 we
+would be done.
+But we also know that
+ǫ ≤
+ν2µ2β2
+22τ+30L4M 3K2τ+2 .
+Then it is enough to check that
+ν2µ2β2
+22τ+30L4M 3K2τ+2 ≤
+µν2βρ1
+22τ+13M 2Kτ+1 .
+And this holds because µ ≤ 27ρ1L4Kτ+1
+– 3 Lastly we want to see that
+ξq−1 ≤ min((Mq − Mq−1)δ(q)
+2
+R , (µq−1 − µq)ρ(q−1)
+2
+).
+Observe that R does not depend on q because at each iteration ˆh(q)
+singular part is not modified. ˆh(q) = ˆh(q) + R(q)
+0
+and R0 is analytic
+depending only on the action coordinates. By induction hypothesis,
+we know that
+ξq−1 = |∂R(q)
+0
+∂I
+|Gq,ρ(q)
+2
+≤ 4MKτ+1ε
+νβ2(τ+2)q .
+We are going to check the two different inequalities separately
+
+76
+6. A NEW KAM THEOREM
+(a) ξq−1 ≤ (Mq − Mq−1) δ(q)
+2
+R . Note that Mq = (2 −
+1
+2q )M, then
+Mq − Mq−1 = M
+2q .
+δ(q)
+2
+≥
+νβ
+64M(K2q−1)τ+1 ≥
+νβ
+64M(K2q)τ+1
+=
+νβ
+64MKτ+1
+1
+2qτ+q .
+We deduce
+(Mq − Mq−1)δ(q)
+2
+≥
+νβ
+64Kτ+1
+1
+2τq+2q .
+Hence we only need to check that
+4MKτ+1ε
+νβ2(τ+2)q ≤
+νβ
+64Kτ+1
+1
+2τq+2q .
+The previous condition holds if and only if
+4MKτ+1ε
+νβ
+≤
+νβ
+26Kτ+1 ⇔ ε ≤
+ν2β2
+2K2τ+2M .
+On the other hand, let us use again that ε ≤
+ν2µ2β2
+22τ+30L4M3K2τ+2 .
+If we apply the condition µ ≤ 2τ+6L2M in the last expression
+we obtain:
+ε ≤ ν2β222τ+12L4M 2
+22τ+30L4M 3K2τ+2 =
+ν2β2
+28K2τ+2M .
+(b) ξq−1 ≤ (µq−1 − µq)ρ(q−1)
+2
+.
+Observe that
+µq = (1 + 1
+2q )µ
+2 ,
+(µq−1 − µq) = ((1 +
+1
+2q−1 ) − (1 + 1
+2q ))µ
+2 = (
+1
+2q−1 − 1
+2q )µ
+2
+=
+�2 − 1
+2q
+� µ
+2 = 1
+2q
+µ
+2 =
+µ
+2q+1
+Also,
+ρ(q−1)
+2
+=
+νβ
+32MKτ+1
+q
+=
+νβ
+32M(K2q−1)τ+1
+≥
+νβ
+32MKτ+12q(τ+1) .
+Then,
+(µq−1 − µq)ρ(q−1)
+2
+≥
+µ
+2q+1
+νβ
+32MKτ+12q(τ+1) .
+Then we only have to check that
+4MKτ+1ε
+νβ2τq+2q−2
+≤
+µ
+2q+1
+νβ
+32MKτ+12q(τ+1)
+=
+µ
+2τq+2q+1
+νβ
+32MKτ+1 .
+Which holds if and only if
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+77
+MKτ+1ε
+νβ2−2
+≤ µ
+2
+νβ
+32MKτ+1.
+Then,
+ε ≤ µ2−2
+2
+ν2β2
+32M 2K2τ+2 =
+µν2β2
+28M 2K2τ+2 .
+But we know
+ε
+≤
+ν2µ2β2
+2τ+30L4M3K2τ+2
+≤
+ν2µ2τ+5L4Mβ2
+2τ+30L4M3K2τ+2
+=
+ν2µβ2
+225M2K2τ+2
+≤
+µν2β2
+28M2K2τ+2
+as we wanted.
+In the second inequality we used that µ ≤
+2τ+5L4M.
+So, finally, we can apply the inductive lemma 6.32 with the pa-
+rameters mentioned previously in this section. Hence we obtain a
+canonical transformation Φ(q) and a transformed hamiltonian H(q) =
+h(q) + R(q). The new domains Gq ⊂ G′
+q−1 are going to be specified
+in the following lines. So now we are going to prove 2a,2b,2c,2d,2e.
+– 2a. We want to see εq := ∥DR(q)∥Gq,ρ(q),cq+1 ≤
+8ε
+νρ12(2τ+2)q.
+By the second result of proposition 6.32 we have:
+(6.20)
+εq ≤ e−Kqδ(q)
+1 εq−1 + 14AqKτ
+q
+β′
+q−1δ(q)
+2
+ε2
+q−1.
+Now we are going to bound each term of the right hand of the
+expression at a time.
+Recall that δ(q)
+1
+≥
+νρ1
+82ν(q−1) .
+Kqδ(q)
+1
+≥
+K2q−1 νρ1
+8 2−ν(q−1)
+=
+νρ1
+8 K2(1−ν)(q−1)
+≥
+12(τ+2)
+8
+2(1−ν)(q−1)
+=
+(3/2τ + 3)2(1 − ν)(q − 1)
+≥
+(2τ + 3) 3
+4 ≥ (2τ + 3) ln 2,
+where we used that K ˆρ ≥ 1 and hence K ≥ 12(τ+2)
+νρ1
+. So we
+conclude that e−Kqδ(q)
+1
+≤
+1
+22τ+3 , and we have bounded the first
+term of 6.20. Let us bind the second one.
+On one hand, we have that
+14AqKτ
+q
+β′
+q−1
+≤ 14 · 5Kτ
+q
+νβ
+4
+≤ 29K2
+q
+νβ
+where we have used that β′
+q ≥ νβ
+4 and Aq ≤ 5.
+
+78
+6. A NEW KAM THEOREM
+Now we are going to apply that εq−1 ≤
+8ε
+νρ12(2τ+2)(q−1) , δ(q)
+2
+≥
+νβ
+64MKτ+1
+q
+and ǫ ≤
+ν3ρ1β2
+22τ+22MK2τ+1 to obtain
+14AqKτ
+q
+β′
+q−1δ(q)
+2 εq−1
+=
+14AqKτ
+q
+β′
+q−1
+1
+δ(q)
+2 εq−1
+≤
+29Kτ
+q
+νβ
+64MKτ+1
+q
+νβ
+8ε
+νρ12(2τ+2)(q−1)
+≤
+218MK2τ+1
+q
+ν3β2ρ12(2τ+2)(q−1)
+ν3ρ1β2
+22τ+22MK2τ+1
+≤
+2182(q−1)(2τ+1)−(2τ+2)(q−1)−(2τ+22)
+=
+2(1−q)2−2τ−4 =
+1
+22τ+32q−1 .
+This gives us the bound of the second term of 6.20. Now we
+put both bounds together:
+εq ≤
+1
+22τ+3 εq−1 +
+1
+22τ+3
+1
+2q−1 εq−1 ≤
+1
+22τ+2 εq���1.
+That implies εq ≤
+ǫ
+2(2τ+2)(q−1) as we wanted. Because we can
+assume νρ1 ≤ 1.
+– 2b
+Let us write σ(q)
+2
+= ρ(q−1)
+2
+− δ(q)
+2 /2 = ρ(q)
+2
++ δ(q)
+2 /2 ≥ ρ(q)
+2 , then
+ηq = |R(q)
+0 |Gq,ρ(q)
+2
+≤ |R(q)
+0 |Gq,σ(q)
+2 .
+By the inductive lemma 6.32:
+ηq
+≤
+7AqKτ
+q
+cqβ′
+q−1 ε2
+q−1
+≤
+7AqKτ
+q
+β′
+q−1 ε2
+q−1
+δ(q)
+1
+δ(q)
+2
+=
+14AqKτ
+q
+β′
+q−1δ(q)
+2 ε2
+q−1
+δ(q)
+1
+2
+≤
+1
+22τ+32q−1 εq−1
+δq)
+1
+2
+≤
+1
+2
+δ(q)
+1
+22τ+32q−1
+ε
+νρ12(2τ+2)(q−1)
+≤
+1
+2
+νρ1
+4·2ν(q−1)
+1
+22τ+32q−1
+8ε
+νρ12(2τ+2)(q−1)
+≤
+ε
+2(2τ+3)q .
+For the second part we only need to apply Cauchy inequalities:
+ξq ≤
+2
+δ(q)
+2
+|Rq
+0|Gq,ρ(q)
+2
+≤
+2
+δ(q)
+2
+ε
+2(2τ+3)q .
+– 2c and 2d are direct from lemma 6.32.
+– 2e We need to consider again the results from lemma 6.32
+with Fq as F. We have to check the condition Fq ⊂ F ′
+q−1 −
+4Mq−1εq−1
+µq
+. Let us define dq := βq−βq−1
+2Kτ+1
+q
+. Using that F ′
+q−1 :=
+(F − βq−1) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k.
+β′
+q−1
+|k|τ
+1 ) we have
+F ′
+q−1 − dq ⊃ (F − (βq−1 + dq)) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k, β′
+q−1
+|k|τ
+1
++ |k|dq).
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+79
+Moreover,
+
+
+
+
+
+
+
+
+
+
+
+βq−1 + dq
+≤
+βq, and
+β′
+q−1
+|k|τ
+1 + |k|dq
+=
+β′
+q−1+|k|τ
+1 |k|
+βq−βq−1
+2Kτ+1
+q
+|k|τ
+1
+≤
+β′
+q−1+Kτ+1
+q
+βq−βq−1
+2Kτ+1
+q
+|k|τ
+1
+=
+β′
+q−1+
+βq
+2 −
+βq−1
+2
+|k|τ
+1
+=
+βq
+|k|τ
+1 .
+Now if we see that 4Mq−1εq−1
+µq
+≤ dq we will have the inclusion
+we want. Observe that 4Mq−1
+µq
+≤ 4·2M
+µ/2 = 16M
+µ . So, it is enough
+to check that 16M
+µ εq−1 ≤ dq.
+εq−1
+≤
+8ε
+νρ12(2τ+2)q
+≤
+8ν3ρ1β2
+νρ12(2τ+2)q2(2τ+22)MK2τ+1
+≤
+8ν2β2
+2(2τ+2)q+2τ+20MK2τ+1
+≤
+8ν2β
+2(βq−βq−1)
+ν
+2(τ+1)q+(τ+1)q+2τ+20MK2τ+1
+=
+8νβ2(βq−βq−1)
+2(τ+1)+(τ+1)q+2τ+20MKτ+1
+q
+Kτ
+=
+8νβ2
+2(τ+1)+(τ+1)q+2τ+19MKτ
+(βq−βq−1)
+2Kτ+1
+q
+=
+νβ
+2(τ+1)+(τ+1)q+2τ+15MKτ dq
+≤
+νβ
+23τ+16MKτ dq.
+Hence, it is enough to prove the following:
+16M
+µ
+νβ
+23τ+16MKτ dq ≤ dq.
+Wich holds if and only if
+16M
+µ
+νβ
+2τ+16MKτ ≤ 1 ⇔ Kτ ≥
+νβ
+µ2τ+12 ,
+which we assumed when choosing K.
+(3) Convergence of diffeomorphisms
+Now we are going to prove the convergence of the successive maps
+u(q) : Gq → Fq
+i.e. we want to see that exist proper sets G∗, F ∗ and an analytical
+map u∗ such that u(q) : Gq → Fq converge to u∗ : G∗ → F ∗.
+Let us use lemma 6.32 as before.
+For q ≥ 1 we obtain
+|u(q) − u(q−1)|Gq ≤ ξq
+and
+|(u(q))−1 − (u(q−1))−1|Fq ≤ εq
+µq
+.
+Now, because the following two inequalities hold
+�
+ξq
+≤
+4MKτ+1ε
+νβ2(τ+2)q
+εq
+µq
+≤
+8ε
+νρ122τ+2q
+1
+(1+ 1
+2q ) µ
+2 =
+8ε2q−1
+νβ2(2τ+2)q(2q+1)µ
+
+80
+6. A NEW KAM THEOREM
+the sequences uq and (u(q))−1 converge to maps u∗ and Υ respectively.
+These maps are defined on the following sets:
+G∗
+:=
+�
+q≥0 Gq,
+F ∗
+:=
+�
+q≥0 Fq = (F − β) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k,
+β
+|k|τ
+1 ).
+The second equality holds because F ∗ is a compact for being the
+intersection of compact sets. We can now deduce that
+|u∗ − u(q)|G∗
+≤
+�
+s≥q |u(q) − u(q−1)|G∗
+≤
+�
+s≥q |u(q) − u(q−1)|G
+≤
+�
+s≥q ξq.
+with the same argument we see that |Υ−(u(q))−1|F ∗ ≤ . . . ≤ �
+s≥q
+εq
+µq .
+The next steps are going to be to prove that Gq ⊂ Gq−1 − 2εq−1
+µq−1 and
+Fq ⊂ Fq−1 − 4Mq−1εq−1
+µq−1
+. If we check it and take the limit we would have:
+G∗ ⊂ Gq −
+�
+s≥q
+2εq
+µq
+and
+F ∗ ⊂ Fq −
+�
+s≥q
+4Mqεq
+µq
+.
+Let us first check Fq ⊂ Fq−1 − 4Mq−1εq−1
+µq−1
+. Let us define x := 4Mq−1
+µq−1 .
+Fq−1 − x
+⊃
+(F − (βq−1 + x)) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k, βq−1
+|k|τq + |k|x)
+⊃
+(F − (βq−1 + x)) \ �
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k,
+βq−1+Kτ+1
+q
+x
+|k|τq
+).
+To have the inclusion we want, we have to check that:
+(a) βq−1 + x ≤ βq.
+(b)
+βq−1+Kτ+1
+q
+x
+|k|τ
+1
+≤
+βq
+|k|τ
+1 ⇔ βq−1 + Kτ+1
+q
+x.
+Since the second one implies the first we will only check the second
+one.
+βq−1 + Kτ+1
+q
+x
+=
+βq−1 + Kτ+1
+q
+4Mq−1εq−1
+µq−1
+≤
+βq−1 + Kτ+1
+q
+16Mεq−1
+µ
+≤
+βq−1 + Kτ+1
+q
+dq
+=
+βq−1 + Kτ+1
+q
+βq−βq−1
+2Kτ+1
+q
+=
+βq−1 − βq−1/2 + βq/2
+=
+βq−1+βq
+2
+=
+βq
+Where we have used that 16Mεq/µ ≤ dq and that βq is monotonically
+increasing with q.
+The inclusion Gq ⊂ Gq−1 − 2εq−1
+µq−1 is given as a result of the lemma
+6.32.
+So we proved what we wanted. We are now going to see that u∗ is
+one-to-one on G∗ and that u∗(G∗) = F ∗.
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+81
+Tale I ∈ G∗, we have that u(q)(I) ∈ Fq for every q. Hence u∗(I) ∈ F ∗,
+and we deduce that u∗(G∗) ⊂ F ∗.
+With the same argument, we see
+Υ(F ∗) ⊂ G∗. Let us prove that Υ(u∗(I)) = I.
+|Υ(u∗(I)) − I|
+≤
+|Υ(u∗(I)) − (u(q))−1(u∗(I))
++(u(q))−1(u∗(I)) − (u(q))−1(u(q)(I))|
+≤
+|Υ(u∗(I)) − (u(q))−1(u∗(I))|
++|(u(q))−1(u∗(I)) − (u(q))−1(u(q)(I))|
+≤
+|Υ − (u(q))−1|F ∗ +
+1
+µq |u∗ − u(q)|G∗.
+Where to bound the second term we used the mean value theorem,
+i.e. |u(q)(x) − u(q)(y)|Gq ≤ | ∂
+∂I u(q)|Gq|x − y|, and the fact that because
+of the µq-non-degeneracy, | ∂u(q)
+∂I | ≥ µq|v|, ∀v ∈ Rn and ∀I′ ∈ Gq. Note
+that we can use the mean value theorem because u∗(I) − u(q)(I) belongs
+to Fq because 4Mqεq
+µq
+≥ ξq. Let us prove this inequality. If we want to see
+4Mqεq
+µq
+≥ ξq, it is enough to see 4Mεq
+µ
+≥ ξq.
+ξq
+≤
+2
+δ(q)
+2 |R(q)
+0 |Gq,σ(q)
+2
+≤
+2
+δ(q)
+2
+δ(q)
+1
+εq−1
+2
+1
+22τ+32q−1
+=
+1
+cq
+1
+22τ+22q−1 εq−1 ≤ 4M
+µ εq−1
+The last inequality holds true if and only if
+µ
+≤
+β2ν(q−1)22τ+32q−1
+Kτ+1
+q
+ρ14
+=
+β2ν(q−1)22τ+32q−1
+Kτ+12(τ+1)(q−1)ρ14
+≤
+β22τ+3
+Kτ+1ρ14
+=
+β22τ+1
+Kτ+1ρ1
+≤
+β22τ+1
+(
+1
+νρ1 )τ+1ρ1 = βντ+122τ+1ρt
+1au
+as we assumed in the statement of the theorem.
+Since the bound
+obtained tends to 0, we have Υ(u∗(I)) = I and hence u∗ is one-to-one.
+Analogously we obtain u∗(Υ(J)) = J
+∀J ∈ F ∗. Finally, u∗ is one-to-one
+and u∗(G∗) = F ∗. Note also that from the inductive lemma we obtain
+|h(q) − h(q−1)|Gq,ρ(q−1)
+2
+≤ ηq−1. Also, observe the following bound that we
+are going to use in the next sections.
+|u∗ − u(q)|G∗ ≤
+�
+s≥q
+4MKτ+1ε
+νβ2(τ+2)s .
+(4) Convergence of the canonical transformations
+Let σ(q) = ρ(q−1) − δ(q)
+2 /2. Observe that this definition implies that
+σ(q) − ρ(q) = δ(q)
+2
+and σ(q) − δ(q)
+2
+= ρ(q).
+Observe that applying the
+inductive lemma 6.32:
+|Φ(q) − id|Gq,σ(q),cq
+≤
+2Aq−1Kτ
+q
+β′
+q−1
+εq−1
+
+82
+6. A NEW KAM THEOREM
+≤
+2·5·4
+νβ
+8ε
+νρ12(2τ+2)(q−1)
+≤
+29Kτ ε
+ν2ρ1β2(τ+2)(q−1)
+≤
+29Kτν3ρ1β2
+ν2ρ1β2(τ+2)(q−1)22τ+22MK2τ+1
+≤
+29νβ
+2(τ+2)(q−1)22τ+20MKτ+1
+=
+νβ
+26M(K2q−1)τ+1
+29
+2(q−1)22τ+14
+≤
+δ(q)
+2
+1
+2(q−1)22τ+5
+≤
+δ(q)
+2
+2(q−1)32,
+where we have used that δ(q)
+2
+≥
+νβ
+84MKqP τ+1, ε ≤
+ν3ρ1β2
+22τ+20MK2τ+1 , β ≤
+8MKτ+1ρ2
+ν
+and β′
+q−1 ≥ νβ
+4 .
+Now, recall that ˆδc = min(cδ1, δ2), then ˆδcq = min(cqδ(q)
+1 , δ(q)
+2 ) =
+min(δ(q)
+2 , δ(q)
+2 ) = δ(q)
+2 .
+Now using that |DΥ|G,ρ−δ,c ≤ |Υ|G,ρ,c
+ˆδc
+, we can obtain:
+|DΦ(q) − Id|Gq,ρ(q),cq
+=
+|D(Φ(q)) − id|Gq,ρ(q),cq
+≤
+|D(Φ(q)) − id|Gq,σ(q)−δ(q)
+2
+,cq
+≤
+|Φ(q)−id|Gq,σ(q),cq
+ˆδcq
+≤
+|Φ(q)−id|Gq,σ(q),cq
+δ(q)
+2
+≤
+2|Φ(q)−id|Gq,σ(q),cq
+δ(q)
+2
+≤
+2
+δ(q)
+2
+δ(q)
+2
+2(q−1)·32 ≤
+1
+2q−116 ≤
+1
+2(q−1)4
+Let x, y be such that the segment joining them is contained in Dρ(q)(Gq).
+Using the mean value theorem one can deduce the following bound:
+|Φq(x) − Φq(y)|cq ≤ |DΦ(q)|Gq,ρ(q),cq · |x − y|cq.
+By 22, in particular |Φ(q)(x)−x|cq ≤ δq
+2 and |Φ(q)(y)−y|cq ≤ δq
+2. Then
+the segment that join Φ(q)(x) and Φ(q)(y) is contained in Dρ(q−1)(Gq−1) =
+Dρ(q)+δ(q), because Gq ⊂ Gq−1 − 2εq−1
+µq−1 and because ρ(q) − ρ(q−1) ≤ δ(q)
+2
+because ρ(q) − ρ(q−1) = δ(q)
+2 .
+Therefore we can apply the mean value theorem once again:
+|Φ(q−1)(Φ(q)(x)) − Φ(q−1)(Φ(q)(y))|cq−1
+≤ |DΦ(q−1)|Gq−1,ρq−1,cq−1|Φ(q)(x) − Φ(q)(y)|cq−1
+≤ 2τ+1−ν|DΦ(q−1)|Gq−1,ρq−1,cq−1|Φ(q)(x) − Φ(q)(y)|cq,
+where we have used that cq−1/cq =
+δ(q−1)
+2
+/δ(q−1)
+1
+δ(q)
+2
+/δ(q)
+1
+=
+δ(q−1)
+2
+δ(q)
+2
+δ(q)
+1
+δ(q−1)
+1
+=
+2τ+1 1
+2ν = 2τ+1−ν.
+Using the previous bounds and iterating by q, we obtain the following:
+|Ψ(q)(x) − Ψ(q)(y)|c1
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+83
+≤ 2(τ+1−ν)(q−1)|DΦ(1)|G1,ρ(1),c1 · . . . · |DΦ(q)|Gq,ρ(q),cq|x − y|cq
+≤ 2(τ+1−ν)(q−1)(1 + 1
+4)(1 +
+1
+4·2) · . . . · (1 +
+1
+4·2q−1 )|x − y|cq
+≤ 2(τ+1−ν)(q−1)e1/2|x − y|cq ≤ 2(τ+1−ν)(q−1) · 2|x − y|cq.
+Which holds for q ≥ 1 and for every x, y such that the segment joining
+them is contained in Dρ(q)(Gq). Now, given q ≥ 2 and x ∈ Dρ(q)(Gq) let
+y = Φ(q)(x):
+|Ψ(q)(x) − Ψ(q−1)(x)|c1
+=
+|Ψ(q−1)(Φ(q)(x)) − Ψ(q−1)(x)|c1
+≤
+2(τ+1+ν)(q−2)2|Φ(q)(x) − x|cq−1
+≤
+2(τ+1+ν)(q−1)2|Φ(q)(x) − x|cq
+≤
+2(τ+1+ν)(q−1)2δ(q)
+2
+≤
+2(τ+1+ν)(q−1)2
+28Kτε
+ν2ρ1β2(τ+2)(q−1)
+=
+29Kτε
+ν2ρ1β2(1+ν)(q−1) .
+Which holds even for q = 1 by setting Ψ(0) = id by 22. Hence 25
+implies that Ψ(q) converges to a map
+Ψ∗ : D(ρ1/4,0)(G∗) = W ρ1
+4 (Tn) × G∗ → Dρ(G).
+And we deduce for every q ≥ 0 that
+|Ψ∗ − Ψ(q)|G∗,( ρ1
+4 ,0),c1 ≤
+210Kτε
+ν2ρ1β2(1+ν)q .
+Moreover by taking the limit to the equation
+H ◦ Ψ(q) = h(q) + R(q)
+we see that H ◦ Ψ∗ = h∗(I) on D( ρ1
+4 ,0)(G∗).
+(5) Stability estimates
+Next we see that for q → ∞, the motions associated to the trans-
+formed hamiltonian ˆH(q) = ˆh(q) + R(q) and the quasi-periodic motions of
+ˆh(q) get closer and closer.
+Let us denote
+� x(q)(t) = (φ(q)(t), I(q)(t))
+the trajectory of H(q),
+ˆx(q)(t) = (ˆφ(q)(t), ˆI(q)(t))
+the trajectory of ˆH(q)
+corresponding to a given initial condition x(q)(0) = x∗
+0 = (φ∗
+0, I∗
+0) ∈
+Tn × Gq. Let
+�
+˜x(q)(t)
+:=
+(˜φ(q)(t), I∗
+0) = (φ∗
+0 + u(q)(I∗
+0))t, I∗
+0 ,
+ˆ˜x(q)(t)
+:=
+(ˆ˜φ(q)(t), I∗
+0) = (φ∗
+0 + u′(q)(I∗
+0))t, I∗
+0
+the corresponding trajectories of the integrable parts of h(q) and ˜h(q)
+respectively. Recall that ˆh(q)(I) = h(q)(I)+ζ(q)(I1) = h(q)(I)+q0 log(I1)+
+�m−1
+i=1 qi 1
+Ii
+1 and u′(q) = ¯Bu(q) + ¯
+A(I1). It is clear that ˜x(q)(t) and ˆ˜x(q)(t)
+are defined for all t ∈ R.
+Let us denote:
+
+84
+6. A NEW KAM THEOREM
+Tq = inf{t > 0 : |I(q)(t)−I∗
+0| > δ(q+1)
+2
+or |φ(q)(t)−˜φ(q)(t)|∞ > δ(q+1)
+1
+}.
+ˆTq = inf{t > 0 : |ˆI(q)(t) − I∗
+0| > δ(q+1)
+2
+or |ˆφ(q)(t) − ˆ˜φ(q)(t)|∞ > δ(q+1)
+1
+}.
+Observe that x(q)(t) and ˆx(q)(t) are defined and belong do Dρ(q)(Gq),
+for 0 ≤ t ≤ Tq and 0 ≤ t ≤ ˆTq respectively, because δ(q) ≤ ρ(q). Also
+recall the Hamiltonian equations. Let us first state the motion equations
+for our Hamiltonian function ˆH(q):
+ιX ˆ
+H(q) ω = d ˆH(q),
+or
+X ˆ
+H(q) = Π(d ˆH(q), ·).
+Let us write
+X ˆ
+H(q) = ˙ˆI(q)
+1
+∂
+∂I1
++ . . . ˙ˆI(q)
+n
+∂
+∂In
++ ˙ˆφ(q)
+1
+∂
+∂φ1
++ . . . + ˙ˆφ(q)
+n
+∂
+∂φn
+.
+Moreover
+d ˆH(q)
+=
+dˆh(q) + dR(q)
+=
+dζ(q) + dh(q) + dR(q)
+=
+�n
+i=1
+∂ζ(q)
+∂Ii +
+n
+�
+i=1
+∂ζ(q)
+∂φi
+�
+��
+�
+=0
++ �n
+i=1
+∂h(q)
+∂Ii
++
+n
+�
+i=1
+∂h(q)
+∂φi
+�
+��
+�
+=0
++ �n
+i=1
+∂R(q)
+∂Ii
++ �n
+i=1
+∂R(q)
+∂φi .
+Recall
+ω =
+
+
+m
+�
+j=1
+cj
+Ij
+1
+
+ dI1 ∧ dφ1 +
+n
+�
+i=2
+dIi ∧ dφi,
+Π =
+1
+��m
+j=1
+cj
+IJ
+1
+� ∂
+∂I1
+∧
+∂
+∂φ1
++
+n
+�
+i=2
+∂
+∂Ii
+∧
+∂
+∂φi
+.
+Then:
+
+
+
+
+
+˙ˆI(q)
+j
+=
+− ∂R(q)
+∂φj (ˆx(q)(t)),
+if j ̸= 1 and
+˙ˆI(q)
+1
+=
+−
+1
+��m
+i=1
+ci
+Ii
+1
+� ∂R(q)
+∂φj (ˆx(q)(t)) = −B(I1) ∂R(q)
+∂φ1 (ˆx(q)(t)).
+Observe that
+(6.21)
+| ˙ˆI(q)
+1 (t)| ≤
+����
+∂R(q)
+∂φ1
+(ˆx(q)(t))
+���� .
+Moreover,
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+85
+
+
+
+
+
+
+
+
+
+
+
+
+
+˙ˆφ(q)
+j
+=
+ˆu(q)
+j (ˆI(q)(t)) + ∂R(q)
+∂Ij (ˆx(q)(t))
+=
+u(q)
+j (ˆI(q)) + ∂R(q)
+∂Ij (ˆx(q)(t))
+if j ̸= 1
+˙ˆφ(q)
+1
+=
+(B(I1)u(q)
+1
++ A(I1))
+�
+��
+�
+u(q)
+1
+(ˆI(q)(t)) + B(I1) ∂R(q)
+∂I1 (ˆx(q)(t)),
+where we have used that ˆu(q)
+j
+= u′(q)
+j
+if j ̸= 1. Using 6.21 we obtain
+| ˙ˆI(q)(t)| ≤
+����
+∂R(q)
+∂φ
+����
+Gq,ρ(q)
+≤ εq.
+Hence,
+| ˙ˆφ(q) − u′(q)(I∗
+0 )|∞
+=
+|u′(q)(ˆI(q)(t)) + ¯B ∂R(q)
+∂I1 (ˆx(q)(t)) − u′(q)(I∗
+0)|∞
+≤
+|u′(q)(ˆI(q))(ˆI(q)(t)) − u′(q)(I∗
+0)|∞ + | ∂R(q)
+∂I (ˆx(q)(t))|∞
+≤
+M ′
+q|ˆI(q)(t) − I∗
+0| + ∥ ∂R(q)
+∂I ∥Fq,ρ(q),∞
+≤
+Mq|ˆI(q)(t) − I∗
+0| +
+εq
+cq+1
+≤
+2Mδ(q+1)
+2
++
+εq
+cq+1 ≤ 3Mδ(q+1)
+2
+.
+Where in the last bound we used that
+(6.22)
+εq
+cq+1
+≤ Mδ(q+1)
+2
+,
+that holds because:
+εq
+cq+1
+≤
+16MKτ+1
+q+1 ρ1
+β2νq
+8ε
+νρ12(2τ+2)q
+≤
+16MKτ+1
+q+1 ρ1
+βeνq
+8
+νρ12(2τ+2)q
+ν2µ2β2
+2τ+30L4M2K2τ+2
+≤
+27Kτ+1
+q+1 νµ2β
+2(2τ+2)q+νq+τ+30L4M2K2τ+2
+≤
+νβµ2
+Kτ+1
+q+1 2νq+τ+23L4M2
+=
+µ2
+2νq+τ+17L4M
+νβ
+26MKτ+1
+q+1
+≤
+µ2
+2τ+17+νqL4M δ(q+1)
+2
+≤
+22τ+12L4M2
+2τ+17+νqL4M δ(q+1)
+2
+≤
+2τ−5−νqδ(q+1)
+2
+≤
+2τ
+25+νq Mδ(q+1)
+2
+≤
+Mδ(q+1)
+2
+if q is large enough.
+Thus, since one of the inequalities defining ˆTq has to be an equality
+for t = Tq we obtain,
+δ(q+1)
+2
+=
+|ˆI(q)(Tq) − I∗
+0| ≤ Tqεq,
+or
+δ(q+1)
+1
+=
+|ˆφ(q)(Tq) − ˆ˜φ(q)(T1)|∞ ≤ Tq3Mδ(q+1)
+2
+.
+Hence, ˆTq ≥ min( δ(q+1)
+2
+εq
+,
+δ(q+1)
+1
+3Mδ(q+1)
+2
+) ≥
+1
+3Mcq+1 , where we used again
+6.22.
+
+86
+6. A NEW KAM THEOREM
+Let us denote T ′
+q :=
+1
+3Mcq+1 , then ˆTq ≥ T ′
+q. This implies
+|ˆx(q)(t) − ˆ˜x(q)(t)|cq+1 ≤ δ(q+1)
+2
+for |t| ≤ T ′
+q.
+Since ˆH(q) = ˆH◦Ψ(q) and Ψ(q) is canonical it turns out that Ψ(q)(ˆx(q)(t))
+is a trajectory of ˆH defined for t ≤ T ′
+q. It is important to observe that
+for q big enough this trajectory remains near the torus Ψ(q)(Tn × {I∗
+0}).
+Moreover T ′
+q tends to infinity when q → ∞.
+(6) Invariant tori
+Assume now that x∗
+0 ∈ Tn × G∗ and let us write
+� x∗(t)
+=
+(φ∗
+0 + u∗(I∗
+0 )t, I∗
+0)
+ˆx∗(t)
+=
+(φ∗
+0 + u′∗(I∗
+0)t, I∗
+0)
+for t ∈ R.
+Note that
+|ˆ˜x(q)(t) − ˆx∗(t)|cq+1
+≤
+cq+1|u′(q)(I∗
+0 ) − u′∗(I∗
+0 )|∞|t|
+≤
+cq+1|u′(q) − u′∗|G∗,∞|t|.
+And observe that if |t| ≤
+δ(q+1)
+1
+|u′(q)−u′∗|G∗,∞ =: T ′′
+q then,
+|ˆ˜x(q)(t) − ˆx∗(t)|cq+1
+≤
+cq+1|u′(q) − u′∗|G∗,∞
+δ(q+1)
+1
+|u′(q)−u′∗|G∗,∞
+≤
+δq+1
+2
+δq+1
+1
+δq+1
+1
+= δq+1
+2
+.
+Observe that
+|u′∗ − u′(q)|G∗ = | ¯Bu∗ + ¯
+A − ¯Bu(q) − ¯
+A|G∗
+= |B(u∗ − u(q))|G∗ ≤ |u∗ − u(q)|G∗,
+close enough to Z.
+Hence the bound obtained for |u∗−u(q)|G∗ also holds for |u′∗−u′(q)|G∗.
+|u′∗ − u′(q)|G∗ ≤
+�
+s≥q
+4MKτ+1ε
+νβ2(τ+2)s ≤ 8MKτ+1ε
+νβ2(τ+2)q .
+Using this bound, we see that T ′′
+q tends to infinity because
+T ′′
+q ≥
+� νρ1
+8 · 2νq
+� � νβ2(τ+2)q
+8MKτ+1ε
+�
+=
+ν2βρ1
+64MKτ+1ε2(τ+2−ν)q.
+Then
+|ˆx(q)(t) − ˆx∗(t)|cq+1 ≤ |ˆx(q)(t) − ˆ˜x(q)(t)|cq+1 + |ˆ˜x(q)(t) − ˆx∗(t)|cq+1 ≤ 2δ(q+1)
+2
+.
+when t ≤ T ′′′
+q := min(T ′
+q, T ′′
+q ).
+Next, we see that the trajectory Ψ(q)(x(q)(t)) is very close to Ψ∗(x∗(t))
+for large values of q. This is true because when |t| ≤ T ′′′
+q .
+|Ψ(q)(ˆx(q) − Ψ∗(ˆx∗(t)))|c1
+≤ |Ψ(q)(ˆx(q)(t)) − Ψ(q)(ˆx∗(t))|c1 + |Ψ(q)(ˆx∗(t)) − Ψ∗(ˆx∗(t))|c1
+≤ 2(τ+1−ν)(q−1) · 2|ˆx(q)(t) − ˆx∗(t)|cq + |Ψ(q) − Ψ∗|G∗,(ρ1/4,0),c1
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+87
+≤ 2(τ+1−ν)(q−1) · 4δ(q+1)
+2
++ |Ψ(q) − Ψ∗|G∗,(ρ1/4,0),c1
+≤ 2(τ+1−ν)(q−1) · 4δ(q+1)
+2
++
+210Kτε
+ν2ρ1β2(1+ν)q
+≤
+c1
+cq+1
+4δ(q+1)
+2
+2(τ+1−ν) +
+210Kτε
+ν2ρ1β2(1+ν)q
+≤
+c14
+2(τ+1−ν)
+δ(q+1)
+1
+δ(q+1)
+2
+δ(q+1)
+2
++
+210Kτ ε
+ν2ρ1β2(1+ν)q
+≤
+c14
+2(τ+1−ν) δ(q+1)
+1
++
+210Kτ ε
+ν2ρ1β2(1+ν)q
+where we used that cq−1/cq = 2τ+1−ν then c1/cq+1 = 2(τ+1−ν)q.
+The bound 28 tends to zero. So we deduce, for every fixed t, Ψ(q)(ˆx(q)(t))
+exits or q large enough and its limit is Φ∗(ˆx∗(t)). This fact and the con-
+tinuity of the flow of ˆH imply that Ψ∗(ˆx∗(t)) is also a trajectory of ˆH,
+which is defined for all t ∈ R.
+This holds for every initial condition x∗
+0 = (φ∗
+0, I∗
+0) ∈ Tn × G∗ for this
+reason Ψ∗(Tn × {I∗
+0}) is an invariant torus of ˆH, with frequency vector
+u′∗(I∗
+0). Observe that the energy on the torus is ˆH(Ψ∗(φ∗
+0, I∗
+0)) = h∗(I∗
+0).
+The preserved invariant tori are completely determined by the trans-
+formed actions I∗
+0 ∈ G∗. We are now going to characterize the preserved
+tori by the original action coordinates.
+First, let us see that u( ˆG) ⊂ F ∗. Recall that:
+∆c,ˆq(k, α) = {J ∈ R such that |k ¯Bu(I) + k ¯
+A| < α},
+ˆG = {I ∈ G − 2γ
+µ such that |k ¯Bu(I) + k ¯
+A| <
+β
+|k|τ
+1
+}.
+With this definition is obvious that if I ∈ ˆG then u(I) is
+β
+|k|τ
+1 , K, c, ˆq-
+non-resonant. Hence u(I) /∈ ∆c,ˆq(k,
+β
+|k|τ
+1 ) for all k ̸= 0. Then u( ˆG) ⊂ F ∗.
+We want to find a correspondence between the invariant tori of ˆh and
+the invariant tori of the perturbed system ˆH = ˆh + R, or in the new
+coordinates ˆh∗.
+Recall
+u′ = ¯Bu + ¯
+A,
+u′∗ = ¯Bu∗ + ¯
+A = (
+1
+�m
+i=1
+ci
+Ii
+1
+u∗
+1 +
+�m
+i=1
+ˆqi
+Ii
+1
+�m
+i=1
+ci
+Ii
+1
+, u∗
+2, . . . , u∗
+n).
+Observe u′∗(0, I2, . . . , In) = ˆqm
+cm =
+1
+K′ the inverse of the modular period,
+hence u′∗ and u′ are not one-to-one at Z because they project the first
+component of u∗ and u to
+1
+K′ .
+Let us define I∗
+0 = (u∗)−1(u(I0)), recall that u and u∗ are indeed
+one-to-one even though u′ and u′∗ are not, so I∗
+0 is properly defined.
+With this definition u∗(I∗
+0) = u(I0) and this implies u′∗(I∗
+0 ) = u′(I0).
+Now, let us define T (φ0, I0) = Ψ∗(φ0, I∗
+0).
+We obtain 6.13 because the set T (Tn × {I0}) is an invariant torus
+of the hamiltonian flow of ˆH with frequency vector u′∗(I∗
+0 ) because Tn ×
+{I∗
+0} is an invariant torus for the hamiltonian flow of ˆh∗. And we have
+seen that u′∗(I∗
+0) = u′(I0). In a nutshell, the original frequencies (of the
+unperturbed system) u(I0) for I0 ∈ ˆG are in F ∗ and hence can be seen
+
+88
+6. A NEW KAM THEOREM
+Dρ(G) = Wρ1(Tn × Vρ1(G))
+Wρ1/4(Tn) × G∗
+Wρ1/4(Tn) × G
+Wρ1/4(Tn) × ˆG
+ˆG
+u( ˆG) ⊂ F
+F ∗
+Ψ∗
+u∗
+i
+i
+π
+u| ˆ
+G
+T
+i
+(u∗)−1
+Figure 2. Diagram of the different maps and sets used in the proof.
+as frequencies of the unperturbed system in the new coordinates u∗(I∗
+0).
+Hence we can conclude that for this I0 ∈ ˆG its new (perturbed) solution
+is also linear in a torus (φ0 + u′∗t, I∗
+0) ∈ Ψ∗(Tn × {I∗
+0}) = T (Tn × {I0}).
+And the new frequency vector u′∗ is such that u′∗ = u′.
+Let us now prove 6.14. Let us write, for (φ0, I∗
+0) ∈ W ρ1
+4 (Tn) × G∗.
+Ψ∗(φ0, I∗
+0) = (φ0 + Ψ∗
+φ(φ0, I∗
+0), I∗
+0 + Ψ∗
+I(φ0, I∗
+0 )).
+And for (φ0, I0) ∈ W ρ1
+4 (Tn)× ˆ
+G.
+T (φ0, I0) = (φ0 + Tφ(φ0, I0), I0 + TI(φ0, I0)).
+Then, for (φ0, I0) ∈ W ρ1
+4 (Tn)× ˆ
+G:
+Tφ(φ0, I0) = Ψ∗
+φ(φ0, I∗
+0),
+and
+TI(φ0, I0) = Ψ∗
+I(φ0, I∗
+0) + I0 − I∗
+0 .
+Let us bound the norms of these terms:
+|Ψ∗
+φ(φ0, I∗
+0)|∞
+≤
+1
+c1 |Ψ∗ − id|G∗,( ρ1
+4 ,0),c1
+≤
+16MKτ+1ρ1
+β
+210Kτ ε
+ν2ρ1β
+≤
+214MK2τ+1ε
+ν2β2
+,
+where we used that c1 ≥
+β
+16MKτ+1ρ1 . Then,
+Ψ∗
+I(φ0, I∗
+o)
+≤
+|Φ∗ − id|G∗,( ρ1
+4 ,0,c1)
+≤
+210kτ ε
+ν2ρ1β .
+Now it only remains the term I∗
+0 − I0:
+|I∗
+0 − I0| ≤ |(u∗)(−1) − (u)(−1)|F ∗ ≤
+�
+s≥0
+ξs
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+89
+≤
+�
+s≥0
+4MKτ+1ε
+νβ2(τ+2)s ≤ 8MKτ+1ε
+νβ2(τ+2) .
+Let us put everything together and use ˆρ ≤ νρ1, K ≤ 2/ˆρ and β =
+γ/L.
+|Ψ∗
+φ(φ0, I∗
+0)|∞
+≤
+214M( 2
+ˆ
+ρ )2τ+1ε
+ν2( γ
+L )2
+≤
+22τ+15ML2
+ν2 ˆρ2τ+1
+ε
+γ2
+|Ψ∗(φ0, I∗
+0)| + |I∗
+0 − I0|
+≤
+210( 2
+ˆ
+ρ )τε
+ν ˆρ( γ
+L )
++
+8M( 2
+ˆ
+ρ )τ+1ε
+ν( γ
+L )2(τ+2)
+=
+210+τ Lε
+ν ˆρτ+1γ +
+8M2τ+1Lε
+ν ˆρτ+1γ2(τ+2)
+≤
+220+τ Lε+M2τ+4Lε
+ν ˆρτ+1γ
+≤ 210+τL(1+M)
+ν ˆρτ+1
+ε
+γ
+(7) Estimate of the measure
+Finally, we carry out the estimate of part 3. Let us write
+ˆG∗ = (u∗)−1(u( ˆG)).
+The invariant tori fill the set
+T (Tn × ˆG) = Ψ∗(Tn × ˆG∗)
+i.e. all the tori inside T (Tn × ˆG) are invariant although there are more of
+them. Because Ψ(q) are hamiltonian transformations, in particular, they
+preserve the volume:
+meas[Ψ(q)(Tn × ˆG∗)] = meas(Tn × ˆG∗) = (2π)nmeas( ˆG∗).
+Now, let us consider the measure of the limit:
+meas[Ψ∗(Tn × ˆG∗)].
+To do this we use the superior limit of sets:
+∞
+�
+n=q
+∞
+�
+j=q
+(Ψ(j)(Tn × ˆG∗)).
+Because Ψ(j)(Tn × ˆG∗) are compact and we have the bound
+|Ψ∗ − Ψ(q)|G∗,( ρ1
+4 ,0),c1 ≤
+210Kτε
+ν2ρ1β2(1+ν)q ,
+�∞
+j=q(Ψ(j)(Tn × ˆG∗)) is also compact. All the measures are well-defined
+and we can say that
+meas[Ψ∗(Tn × ˆG∗)] ≥ (2π)nmeas( ˆG∗).
+Then, to bound the measure of the complement of the invariant set it
+is enough to bound the measure of G \ ˆG∗.
+But first, we are going to define some auxiliary sets. Let ˜β = 2γM
+µ ,
+˜βq = (1−
+1
+2νq )˜β. Note that ˜β ≥ β if and only if µ ≤ 2ML and we assumed
+µ ≤ 2τ+6L2M.
+Then, for q ≥ 0 we define
+
+90
+6. A NEW KAM THEOREM
+˜Fq = (F − ˜βq) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k,
+˜βq
+|k|τ
+1
+),
+˜Gq = (u(q))−1( ˜Fq)
+and
+˜F ∗ =
+�
+q≥0
+˜Fq = (F − ˜β) \
+�
+k∈Zn\{0}
+|k|1≤K
+∆c,ˆq(k,
+˜β
+|k|τ
+1
+),
+˜G∗ =
+�
+q≥0
+˜Gq.
+In order to prove the bounds, we need to prove previously the inclu-
+sions ˜G∗ ⊂ ˆG∗ and ˜G0 ⊂ G.
+(a) G ⊃ ˜G0 = (u(0))−1( ˜F0) = (u)−1(F − ˜β), but we know u(G) = F.
+(b) ˜G∗ ⊂ ˆG∗. Take I ∈ ˜G∗, then I ∈ ˜Gq∀q ≥ 0. Hence ∃J ∈˜˜Fq∀q such
+that u(q)(I). Then ∃J ∈ ˜F ∗ such that u∗(J) = I.
+If we check that J ∈ u( ˜G) we obtain (u∗)−1(J) = I ∈ ˆG∗ and we will
+be done. We want ˜F ∗ ⊂ u( ˆG). Because we are taking out all the
+resonances in ˜F ∗ it is enough to see (F − ˜β) ⊂ u(G − 2γ
+µ ). We only
+need to use that | ∂u
+∂I |G,ρ2 ≤ M. Then F − ˜β ⊂ u(G − 2γ
+µ ). This holds
+if and only if
+˜β
+M ≤ 2γ
+µ which is true because ˆβ ≤ 2γM
+µ .
+Then, we proceed as follows
+meas(G \ ˆG∗)
+≤
+meas(G \ ˜G∗)
+≤
+meas( ˜G0 \ ˜G∗)
+≤
+�∞
+q=1 meas( ˜Gq−1 \ ˜Gq).
+For q ≥ 1 we obtain the following estimate:
+meas( ˜Gq−1 \ ˜Gq) ≤
+1
+| det( ∂u(q−1)
+∂I
+(I))|
+meas(
+u(q−1)( ˜
+Gq−1)
+� �� �
+˜Fq−1
+\(
+u(q−1)( ˜
+Gq)
+�
+��
+�
+˜Fq − εq−1)).
+Where we have used lemma 6.32. Also det( ∂u(q−1)
+∂I
+(I)) ≥ µn
+q−1 because
+of the µq−1-non-degeneracy condition all the eigenvalues have to be greater
+than µq−1.
+meas( ˜Gq−1 \ ˜Gq)
+≤
+1
+µn
+q−1 meas( ˜Fq−1 \ ( ˜Fq − εq−1))
+≤
+2n
+µn meas( ˜Fq−1 \ ( ˜Fq − εq−1)).
+Now, we are going to apply lemma 6.30 with
+˜Fq−1 = F(˜βq−1, ˜βq−1, Kq − 1)
+and ˜Fq = F(˜βq, ˜βq, Kq).
+Applying the lemma:
+
+6.3.
+A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS
+91
+meas( ˜Fq−1 \ ˜Fq) ≤ D(˜βq − ˜βq−1)
++2(diamF)n−1
+
+
+
+
+�
+k∈Zn\{0}
+|k|1≤Kq−1
+˜βq − ˜βq−1
+|k|τ
+1|k|2,ω
++
+�
+k∈Zn\{0}
+Kq−1≤|k|1≤Kq
+˜βq
+|k|τ
+1|k|2,ω
+
+
+
+
+and
+meas( ˜Fq \ ( ˜Fq − εq)) ≤ (D + 2n+1(diamF)n−1Kn)εq.
+Putting everything together (and using that ˜β0 = 0), we get
+(6.23)
+meas(G \ ˆG∗)
+≤
+2n
+µn
+
+D ˜β + 2(diamF)n−1
+�
+k∈Zn\{0}
+˜β
+|k|τ
+1|k|2,ω
++D
+∞
+�
+q=1
+εq−1 + 2n+1(diamF)n−1
+∞
+�
+q=1
+Kn
+q εq−1
+�
+.
+We now only have to check that the series in the previous expression
+converge. Let us check that they converge at Z first and then outside of
+Z. Recall that at Z we take the vectors ¯k ̸= 0.
+�
+k∈Zn\{0}
+¯k̸=0
+1
+|k|τ
+1 |k|2,ω
+≤
+�
+k∈Zn\{0}
+¯k̸=0
+1
+|k|τ
+1 |¯k|
+≤
+�
+¯k∈Zn−1\{0}
+¯k̸=0
+�
+kn∈Z
+√n
+(|¯k|1+|kn|)τ|¯k|1
+≤
+√n2n−1 �∞
+j=1
+�
+kn∈Z
+jn−3
+j+|kn|)τ
+where we used that the number of vectors ¯k ∈ Zn−1 with |¯k|1 = j ≥ 1
+can be bounded by 2n−1jn−2. This series can be bounded by comparing
+it to an integral:
+�
+kn∈Z
+1
+(j + |kn|)τ ≤ 1
+jτ + 2
+� ∞
+0
+dx
+(j + x)τ
+= 1
+jτ +
+2
+(τ + 1)jτ−1 ≤ τ + 1
+τ − 1
+1
+jτ+1 .
+Where we used that τ > 1 because n ≥ 2. Then
+�
+k∈Zn\{0}
+¯k̸=0
+1
+|k|τ
+1|k|2,ω
+≤
+√n2n−1(τ + 1)
+τ − 1
+∞
+�
+j=1
+1
+jτ−n+2
+which converges by the condition τ > n − 1.
+Now let us check that it converges outside of Z.
+�
+k∈Zn\{0}
+1
+|k|τ
+1 |k|2,ω
+=
+�
+k∈Zn\{0}
+¯k̸=0
+1
+|k|τ
+1 |k|2,ω + �
+k∈Zn\{0}
+¯k=0
+1
+|k|τ
+1 |k|2,ω
+=
+�
+k∈Zn\{0}
+¯k̸=0
+1
+|k|τ
+1 |¯k| + �
+k1∈Z
+1
+|k|τ
+1 |k2
+1B(I1)2|
+
+92
+6. A NEW KAM THEOREM
+We have seen before that the first term converges. The second term:
+�
+k1∈Z
+1
+|k1|τ|kτ
+1B(I1)2| =
+1
+B(I1)2
+�
+k1∈Z
+1
+kτ+2
+1
+,
+which converges ∀I1 ̸= 0, i.e. outside of Z.
+Now we go back to the expression 6.23.
+The other terms of that
+expression can be bounded simultaneously inside and outside Z. Now we
+only have to check that the third series converges, because if the third
+converges so does the second. We only have to check that �∞
+q=1 Kn
+q εq−1
+converges. We will use that ε ≤
+8ε
+νρ12(2τ+2)(q−1) .
+�∞
+q=1 Kn
+q εq−1
+=
+Kn �∞
+q=1 2n(q−1)εq−1
+=
+Kn �∞
+q=1
+8ε2n(q−1)
+νρ12(2τ+2)(q−1)
+=
+Kn 8ε
+νρ1
+�∞
+q=1
+1
+2(2τ+2−n)(q−1) .
+Which converges if and only if 2τ + 2 − n ≥ 1. And we are done
+because 2τ ≥ n − 1 since τ ≥ n − 1 by hypothesis.
+Putting everything together:
+meas(G \ ˆG∗)
+≤ 2n
+µn
+
+D2 2γM
+µ
++ 2(diamF)n−1 2γM
+µ
+√n2n−1(τ + 1)
+τ − 1
+∞
+�
+j=1
+1
+jτ−n+2
+D 8ε
+νρ1
+∞
+�
+q=1
+1
+2(2τ + 2)(q − 1)
++2n+1(diamF)n−1Kn 8ε
+νρ1
+∞
+�
+q=1
+1
+2(2τ+2−n)(q−1)
+�
+Now using that
+ε ≤
+ν2µ2β2
+2τ+30L4M 3K2τ+1 ≤ 2τ−18 · 8MKτ+1ρ2
+LMK2τ+2
+γ ≤ 2τ−15ρ2
+LKτ+1 γ
+We can write meas(G \ ˆG∗) ≤ C′γ where C′ depends only on n, µ, D,
+diamF, M, τ, ρ1, ρ2, L, K and if we efine C = (2π)nC′. Hence,
+meas[(Tn × G) \ T (Tn ˆG)] ≤ Cγ.
+□
+
+CHAPTER 7
+Desingularization of bm-integrable systems
+In this chapter, we follow [GMW17], for the definition of the desingularization
+of the bm-symplectic form.
+Definition 7.1. The fǫ-desingularization ωǫ form of ω = dx
+xm ∧
+��m−1
+i=0 xiαm−i
+�
++
+β is:
+ωǫ = dfǫ ∧
+�m−1
+�
+i=0
+xiαm−i
+�
++ β.
+Where in the even case, fǫ(x) is defined as ǫ−(2k−1)f(x/ǫ). And f ∈ C∞(R) is an
+odd smooth function satisfying f ′(x) > 0 for all x ∈ [−1, 1] and satisfying outside
+that
+(7.1)
+f(x) =
+�
+−1
+(2k−1)x2k−1 − 2
+for
+x < −1,
+−1
+(2k−1)x2k−1 + 2
+for
+x > 1.
+And in the odd case, fǫ(x) = ǫ−(2k)f(x/ǫ). And f ∈ C∞(R) is an even smooth
+positive function which satisfies: f ′(x) < 0 if x < 0, f(x) = −x2 + 2 for x ∈ [−1, 1],
+and
+(7.2)
+f(x) =
+�
+−1
+(2k+2)x2k+2 − 2
+if k > 0, x ∈ R \ [−2, 2]
+log(|x|)
+if k = 0, x ∈ R \ [−2, 2].
+Remark 7.2. With the previous definition, we obtain smooth symplectic (in
+the even case) or smooth folded symplectic (in the odd case) forms that agree
+outside an ǫ-neighbourhood with the original bm-forms. Moreover, there is a con-
+vergence result in terms of m. See [GMPS17] for the details.
+To simplify notation, we introduce F m−i
+ǫ
+(x) = ( d
+dxfǫ(x))xi, and hence F i
+ǫ(x) =
+( d
+dxfǫ(x))xm−i. With this notation the desingularization ωǫ is written:
+ωǫ =
+m−1
+�
+i=0
+F m−i
+ǫ
+(x)dx ∧ αm−i + β.
+Definition 7.3. The desingularization for (M, ω, µ) is the triple (M, ωε, µǫ)
+where ωε is defined as above and µε is:
+µ �→ µǫ =
+�
+f1ǫ =
+m
+�
+i=1
+ˆciGi
+ǫ(x), f2(˜I, ˜φ), . . . , fn(˜I, ˜φ)
+�
+,
+where
+µ =
+�
+f1 = c0 log(x) +
+m−1
+�
+i=1
+ci
+1
+xi , f2(I, φ) . . . , fn(I, φ)
+�
+93
+
+94
+7. DESINGULARIZATION OF bm-INTEGRABLE SYSTEMS
+Gi
+ǫ(x) =
+� x
+0
+F i
+ǫ(τ)dτ,
+and ˆc1 = c0 and ˆci−1 = −ici if i ̸= 0. Also
+
+
+
+
+
+
+
+
+
+
+
+˜I = (˜I1, I2, . . . , In),
+˜I1
+=
+� I1
+0
+��m
+i=1 KˆciF i
+ε(τ)
+�m
+i=1
+Kˆcj
+τj
+�
+dτ
+˜φ1 = (˜φ1, φ2, . . . , φn),
+˜φ1
+=
+
+
+�m
+i=1 K��ciF i
+ε(I1)
+�m
+i=1
+Kˆcj
+Ij
+1
+
+ φ1
+Remark 7.4. Observe that with the last definition, when ǫ tends to 0, µǫ tends
+to µ.
+Theorem C. The desingularization transforms a bm-integrable system into an
+integrable system for m even on a symplectic manifold. For m odd the desingular-
+ization transforms it into a folded integrable system. The integrable systems are
+such that:
+Xω
+fj = Xωǫ
+fjǫ.
+Proof. Let us first check the singular part, i.e. let us check that that Xω
+f1 =
+Xωǫ
+f1ǫ. Let us compute the two equations that define each one of the vector fields.
+We have to impose −df1 = ιXω
+f1 ω and −df1ǫ = ιXωǫ
+f1ǫ ωǫ. But observe first that we
+can rewrite ω = �m
+i=1
+1
+xi dx ∧ αi + β and ωǫ = �m
+i=1 F i
+ǫdx ∧ αi + β. The conditions
+translate as:
+−
+m
+�
+i=1
+ˆci
+1
+xi dx = ιXω
+f1
+� m
+�
+i=1
+1
+xi dx ∧ αi + β
+�
+,
+−
+m−1
+�
+i=0
+ˆciF i
+ǫ(x)dx = ιXωǫ
+f1ǫ
+�m−1
+�
+i=0
+F i
+ǫ(x)dx ∧ αi + β
+�
+.
+Since the toric action leaves the form ω invariant, in particular, the singular
+set is invariant, and then Xωǫ
+f1ǫ and Xω
+f1 are in the kernel of dx. Moreover, since β
+is a symplectic form in each leaf of the foliation and Xωǫ
+f1ǫ and Xω
+f1 are transversal
+to this foliation, they are also in the kernel of β.
+−
+m−1
+�
+i=0
+ˆci
+1
+xi dx =
+m−1
+�
+i=0
+1
+xi dx ∧ αi(Xω
+f1),
+−
+m−1
+�
+i=0
+ˆciF i
+ǫ(x)dx =
+m−1
+�
+i=0
+F i
+ǫ(x)dx ∧ αi(Xωǫ
+f1ǫ).
+Then, the conditions over Xω
+f1 and Xωǫ
+f1ǫ are respectively:
+−ˆci = αi(Xω
+f1),
+−ˆci = αi(Xωǫ
+f1ǫ).
+Then, the two vector fields have to be the same.
+Let us now see Xω
+fj = Xωǫ
+fjǫ for j > 1. Assume now we have the bm-symplectic
+form in action-angle coordinates ω = �m
+i=1
+Kˆci
+Ii
+1 dI1 ∧ dφ1 + �n
+i=1 dIi ∧ dφi.
+The differential of the functions are
+
+7. DESINGULARIZATION OF bm-INTEGRABLE SYSTEMS
+95
+df ε
+i
+=
+∂f ε
+i
+∂I1 dI1 + ∂f ε
+i
+∂φ1 dφ1 + �n
+j=2
+�
+∂f ε
+i
+∂Ij dIj + ∂f ε
+i
+∂φj dφj
+�
+=
+∂fi
+∂I1
+��m
+i=1 KˆciF i
+ε(τ)
+�m
+i=1
+Kˆcj
+τj
+�
+dI1 + ∂fi
+∂φ1
+��m
+i=1 KˆciF i
+ε(τ)
+�m
+i=1
+Kˆcj
+τj
+�
+dφ1
++ �n
+j=2
+�
+∂f ε
+i
+∂Ij dIj + ∂f ε
+i
+∂φj dφj
+�
+.
+On the other hand, the desingularized form is:
+ωε =
+m
+�
+j=1
+KˆciF j
+ε (I1)dI1 ∧ dφ1 +
+m
+�
+j=2
+dIj ∧ dφj.
+Hence, one can see that the expression for both Xω
+fj and Xωǫ
+fjǫ is
+Xω
+fj = Xωǫ
+fjǫ =
+∂fi
+∂I1
+�m
+i=1
+Kˆci
+Ii
+1
+∂
+∂φ1
+−
+∂fi
+∂φ1
+�m
+i=1
+Kˆci
+Ii
+1
+∂
+∂I1
++
+n
+�
+j=2
+�∂f ε
+i
+∂Ij
+dIj + ∂f ε
+i
+∂φj
+dφj
+�
+□
+Remark 7.5. The previous lemma tells us that the dynamics of the desingu-
+larized system are identical to the dynamics of the original bm-integrable system in
+the bm-symplectic manifold.
+Hence the desingularized bm-form goes to a folded symplectic form in the case
+m = 2k + 1 and to symplectic for m = 2k. And the bm-integrable system goes to
+a folded integrable system (see [CM22]) in the case m = 2k + 1 and to a standard
+integrable system for n = 2k.
+
+
+CHAPTER 8
+Desingularization of the KAM theorem on
+bm-symplectic manifolds
+The idea of this section is to recover some version of the classical KAM theorem
+by “desingularizing the bm-KAM theorem”, as well as a new version of a KAM
+theorem that works for folded symplectic forms. Observe that no KAM theorem
+is known for folded symplectic forms. The best that is known is a KAM theorem
+for presymplectic structures that was done in [AdlL12]. Desingularizing the KAM
+means applying the bm-KAM in the bm-manifold and then translating the result to
+the desingularized setting.
+To be able to obtain proper desingularized theorems we need to identify which
+integrable systems can be obtained as a desingularization of a bm-integrable system.
+To simplify computations we are going to use a particular case of bm-integrable
+systems, where f1 =
+1
+Im−1
+1
+. We call these systems simple. Observe that by taking
+a particular case of bm-integrable systems we will not get all the systems that can
+be obtained by desingularizing a bm-integrable system, but some of them.
+(1) Even case m = 2k.
+F = (f1 =
+1
+I2k−1
+1
+, f2, . . . , fn), ω =
+1
+Im
+1 dI1 ∧ dφ1 + �n
+j=1 dIj ∧ dφj.
+Observe that close to Z in the even case we can assume f(I1) = cI1
+for some c > (2 −
+1
+22k−1 ). Then fε(I1) =
+1
+ε(2k−1)
+cI1
+ε
+= c′I, hence ωε =
+c′dI1 ∧ dφ1 + �n
+j=1 dIj ∧ dφj. Also F m
+ε (I1) = c′, Gm
+ε (I1) = c′I1. Then,
+� ˜I1
+=
+� I1
+0
+c′
+1/τ m dτ =
+� I1
+0 c′τ mdτ = c′ Im+1
+1
+m+1 ,
+˜φ1
+=
+c′
+1/Im
+1 φ1 = c′Im
+1 φ1
+(8.1)
+F ε = ((m − 1)cm−1c′I1, f2(˜I, ˜φ), . . . fn(˜I, ˜φ)).
+Hence, the systems in this form can be viewed as a desingularization
+of a bm-integrable system.
+Theorem D (Desingularized KAM for symplectic manifolds). Con-
+sider a neighborhood of a Liouville torus of an integrable system Fε as
+in 8.1 of a symplectic manifold (M, ωε) semilocally endowed with coor-
+dinates (I, φ), where φ are the angular coordinates of the torus, with
+ωε = c′dI1∧dφi+�n
+j=1 dIj∧dφj. Let H = (m−1)cm−1c′I1+h(˜I)+R(˜I, ˜φ)
+be a nearly integrable system where
+�
+˜I1
+=
+c′ Im+1
+1
+m+1 ,
+˜φ1
+=
+c′Im
+1 φ1,
+97
+
+98
+8. bm-KAM DESINGULARIZATION
+and
+� ˜I
+=
+(˜I1, I2, . . . , In),
+˜φ
+=
+(˜φ1, φ2, . . . , φn).
+Then the results for the bm-KAM theorem 6.3 applied to Hsing =
+1
+I2k−1
+1
++
+h(I) + R(I, φ) hold for this desingularized system.
+Remark 8.1. This theorem is not as general as the standard KAM,
+but we also know extra information about the dynamics. For instance,
+the perturbation of trajectories in tori inside of Z will be trajectories lying
+inside of Z. In this sense, the theorem is new because it leaves invariant
+an hypersurface of the manifold.
+(2) Odd case m = 2k + 1.
+F = (f1 =
+1
+I2k
+1 , f2, . . . , fn) and ω =
+1
+I2k+1
+1
+dI1 ∧ dφ1 + �n
+j=1 dIj ∧ dφj.
+Before continuing we need the following notions defined in [CM22].
+Definition 8.2. A function f : M → R in a folded symplectic mani-
+fold (M, ω) is folded if df|Z(v) = 0 for all v ∈ V = kerω|Z.
+Definition 8.3. An integrable system in a folded symplectic manifold
+(M, ω) with critical surface Z is a set of functions F = (f1, . . . , fn) such
+that they define Hamiltonian vector fields which are independent (df1 ∧
+. . . ∧ dfn ̸= 0 in the folded cotangent bundle) on a dense subset of Z and
+M, and commute with respect to ω.
+Note that we need to prove that the desingularized functions in this
+case are folded.
+Observe that close to Z in the odd case we can assume f(I1) = −I2
+1+2.
+Then fε(I1) = ε−(2k)f( I1
+ε ) =
+1
+ε2k (−( I1
+ε )2 + 2) = cI2
+1 +
+2
+ε2k . Then
+ωε = 2cI1dI1 ∧ dφ1 +
+n
+�
+j=1
+dIj ∧ dφj.
+Also F m
+ε (I1) = 2cI1, Gm
+ε (I1) = cI2
+1. Then,
+�
+˜I1
+=
+� I1
+0
+2cτ
+1/τ m dτ = 2c I(m+2)
+1
+(m+2) ,
+˜φ1
+=
+2cIm+1
+1
+φ1
+Then the desingularized moment map becomes
+(8.2)
+F ε = ((m − 1)cm−1cI2
+1, f2(˜I, ˜φ), . . . fn(˜I, ˜φ)).
+It is a simple computation to check that these functions are actually
+folded and hence they form a folded integrable system. Note that the
+systems of the form 8.2 can be viewed as a desingularization of a bm-
+integrable system. Then, as we proceeded in the even case:
+Theorem E (Desingularized KAM for folded symplectic manifolds).
+Consider a neighborhood of a Liouville torus of an integrable system Fε as
+in 8.2 of a folded symplectic manifold (M, ωε) semilocally endowed with
+coordinates (I, φ), where φ are the angular coordinates of the Torus, with
+
+8. bm-KAM DESINGULARIZATION
+99
+ωε = 2cI1dI1 ∧ dφ1 + �m
+j=2 dIj ∧ dφj. Let H = (m − 1)cm−1cI2
+1 + h(˜I) +
+R(˜I, ˜φ) a nearly integrable system with
+�
+˜I1
+=
+2c Im+2
+1
+m+2 ,
+˜φ1
+=
+2cIm+1
+1
+φ1,
+and
+� ˜I
+=
+(˜I1, I2, . . . , In),
+˜φ
+=
+(˜φ1, φ2, . . . , φn).
+Then the results for the bm-KAM theorem 6.3 applied to Hsing =
+1
+I2k
+1
++
+h(I) + R(I, φ) hold for this desingularized system.
+Remark 8.4. The last two theorems can be improved if we consider
+bm-integrable systems not necessarily simple.
+
+
+CHAPTER 9
+Potential applications to Celestial mechanics
+All the theory developed in this monograph would not be fertile if we could not
+envisage applications of perturbation theory to actual physical systems. We provide
+several examples from Celestial mechanics and conclude with potential applications
+of our KAM theory to detect periodic trajectories.
+In this chapter we present several examples appearing in Celestial Mechanics
+where singular symplectic forms show up. Some of these examples are contained
+in [MDD+19].
+Most of the singularities appear as a consequence of applying
+regularization techniques. We invite the reader to consult the book [Kna18] for a
+pedagogical approach to the study of regularization.
+This list of examples is of special relevance for this booklet as the theoretical
+results that we obtain such as action-angle coordinates or KAM can be, de facto,
+applied to the list of problems considered below.
+Structures that are symplectic almost everywhere can arise as the result of
+changes of coordinates which do not preserve the canonical symplectic structure.
+For instance: For the Kepler problem given a configuration space R2 and phase
+space T ∗R2, the traditional (canonical) Levi-Civita transformation described as
+follows: identify R2 ∼= C so that T ∗R2 ∼= T ∗C ∼= C2 and treat (q, p) as complex
+variables (q1 + iq2 := u, p1 + ip2 := v). Take the following change of coordinates
+(q, p) = (u2/2, v/¯u), where ¯u denotes the complex conjugation of u. The resulting
+coordinate change can easily be seen to preserve the canonical symplectic form.
+However, this canonical change of coordinates can make the Hamiltonian equations
+more complicated making more difficult to study the dynamics of the system. This
+is why it is often interesting to consider other changes of coordinates where the
+symplectic form is not preserved. Some of them induce new singular forms where
+our geometrical and dynamical techniques can be applied.
+Other examples are discussed in [DKM17].
+9.1. The Kepler Problem
+In suitable coordinates in T ∗ �
+R2 \ {0}
+�
+, the Kepler problem has Hamiltonian
+(9.1)
+H(q, p) = ∥p∥2
+2
+−
+1
+∥q∥.
+With the canonical Levi-Civita transformation (q, p) = (u2/2, v/¯u), this expression
+becomes
+(9.2)
+H(u, v) = ∥v∥2
+2∥¯u∥2 −
+1
+∥u∥2 .
+101
+
+102
+9.
+POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS
+Changes of coordinates preserving the canonical symplectic form leads to more
+complicated equations. So we propose a new reciipe: leave the momentum un-
+changed and examine the transformation (q, p) = (u2/2, p) instead. This can result
+in a simpler Hamiltonian. The transformation is not a symplectomorphism and the
+symplectic form on T ∗R2 pulls-back under the transformation to a two-form which
+is symplectic almost everywhere, but degenerates on a hypersurface of T ∗R2
+Namely, the Liouville one-form p1dq1 + p2dq2 = ℜ(pd¯q) pulls back to
+θ = ℜ
+�
+pd
+� ¯u2
+2
+��
+=
+ℜ (p¯ud¯u)
+=
+p1(u1du1 − u2du2) + p2(u2du1 + u1du2)
+and the associated 2-form −dθ yields a form that is almost everywhere symplectic
+ω = u1du1 ∧ dp1 − u2du1 ∧ dp2 + u2du2 ∧ dp1 + u1du2 ∧ dp2.
+In order to test the nature of this form we wedge the form with itself and we
+find
+ω ∧ ω = (u2
+1 − u2
+2)du1 ∧ dp1 ∧ du2 ∧ dp2
+which is degenerate along the hypersurface given by u1 = ±u2.
+We now consider the restriction of the form to the critical set. It does not have
+maximal rank so it is not a folded symplectic structure. This form is degenerately
+folded and the folding hypersurface is not regular and is described by the equations
+u1 = ±u2.
+9.2. The Problem of Two Fixed Centers
+We now regularize the problem of two fixed centers.
+The problem of two fixed centers is associated to the motion of a satellite moving
+in a gravitational potential generated by two fixed massive bodies. We assume also
+that the motion of the satellite is restricted to the plane in R3 containing the two
+massive bodies.
+The Hamiltonian function in suitable coordinates reads:
+(9.3)
+H = p2
+2m − µ
+r1
+− 1 − µ
+r2
+where µ is the mass ratio of the two bodies (i.e. µ =
+m1
+m1+m2 ).
+The integrability of this problem was first proved by Euler via elliptic coordi-
+nates, where the coordinate lines are confocal ellipses and hyperbola.
+Explicitly, consider a coordinate system in which the two centers are placed at
+(±1, 0), in which the (Cartesian) coordinates are given by (q1, q2). Then the elliptic
+coordinates of the system are given by
+q1 = sinh λ cos ν
+(9.4)
+q2 = cosh λ sin ν
+(9.5)
+for (λ, ν) ∈ R × S1. Thus lines of λ = c and ν = c are given by confocal hyperbola
+and ellipses in the plane, respectively. Similar to the Levi-Civita transformation
+this results in a double-branched covering with branch points at the centers of
+attraction.
+Pulling back the canonical symplectic structure ω = dq ∧ dp we find
+(9.6)
+ω = cosh λ cos ν(dλ ∧ dp1 + dν ∧ dp2) − sinh λ sin ν(dν ∧ dp1 + dλ ∧ dp2)
+
+9.3. DOUBLE COLLISION AND MCGEHEE COORDINATES
+103
+which is degenerate along the hypersurface (λ, ν) satisfying cosh λ cos ν = sinh λ sin λ.
+9.3. Double Collision and McGehee coordinates
+In this section, we describe another example of b-symplectic structure appearing
+quite naturally in physical dynamical systems. From this example, it would seem
+natural that a collection of different examples for bm-symplectic models or even
+bm-folded models would follow.
+But one finds a major problem while pursuing
+these examples. Understanding why this example does not extend to construct
+bm-symplectic models of bm-folded for any m gives a general pattern.
+First let us introduce the McGehee coordinate change for the problem of double
+collision.
+The system of two particles moving under the influence of the generalized po-
+tential U(x) = −|x|−α, α > 0, where |x| is the distance between the two particles,
+is studied by McGehee in [McG81]. We fix the center of mass at the origin and
+hence can simplify the problem to the one of a single particle moving in a central
+force field.
+The equation of motion can be written as,
+(9.7)
+¨x = −∇U(x) = −α|x|−α−2x
+where the dot represents the derivative with respect to time. In the Hamiltonian
+formalism, this equation becomes
+(9.8)
+˙x
+=
+y,
+˙y
+=
+−α|x|−α−2x.
+To study the behavior of this system, the following change of coordinates is sug-
+gested in [McG81]:
+(9.9)
+x
+=
+rγeiθ,
+y
+=
+r−βγ(v + iw)eiθ
+where the parameters β and γ are related with α as follows:
+(9.10)
+β
+=
+α/2,
+γ
+=
+1/(1 + β).
+Identifying the plane R2 with the complex plane C, we can write the symplectic
+form of this problem as ω = ℜ(dx ∧ dy).
+Remark 9.1. To check that a form ω is actually a bm-symplectic form, it is not
+enough to check that the multi-vector field dual to ω∧ω is a section of �2n(bmT M)
+which is transverse to the zero section. One has to check additionally that the
+Poisson structure dual to ω itself is a proper section of �2(bmT M).
+Proposition 9.2. Under the coordinate change (9.9), the symplectic form ω
+is sent to a b-symplectic structure for α = 2.
+Proof. The proof of this proposition is a straightforward computation. Ob-
+serve that the change is not a smooth change, so we are not working with standard
+De Rham forms. But, at the end of the computation it will become clear that the
+form is a b-symplectic form and hence the computations are legitimate. If one does
+
+104
+9.
+POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS
+the change of variables, we obtain:
+(9.11)
+¯y
+=
+rβγ(v − iw)e−iθ.
+dx
+=
+γrγ−1eiθdr + rγeiθidθ.
+d¯y
+=
+r−βγ−1(−βγ)(v − iw)e−iθdr + r−βγe−iθdv
++rβγ(v − iw)e−iθ(−i)dθ.
+By wedging the previous two forms, we obtain:
+(9.12)
+dx ∧ d¯y
+=
+dr ∧ dv(γrγ−1−βγ)
++
+dr ∧ dw(γrγ−1−βγ)
++
+dr ∧ dθ(γrγ−1−βγ(−iv − w))
++
+dθ ∧ dr(irγ−1−βγ(−βγ)(v − iw))
++
+dθ ∧ dv(irγ−βγ)
++
+dθ ∧ dw(irγ−βγ(−i)).
+Now we can take the real part of this form and use that γ − 1 − βγ = −αγ. In the
+new coordinates, the form reads.
+(9.13)
+ω = ℜ(dx ∧ d¯y)
+=
+γr−βγ+γ−1dr ∧ dv − γ(1 − β)r−βγ+γ−1wdr ∧ dθ
+−
+r−βγ+γdw ∧ dθ.
+Moreover, we can use that γ(1 + β) = 1 to simplify the previous expression further
+to:
+(9.14)
+ω = (dr ∧ dv + dr ∧ dw)γr−αγ + dr ∧ dθ(wr−αγ) + dθ ∧ dw(r−αγ+1).
+In order to classify this structure, we wedge it with itself and look at the structure
+of the form in the singular set. Wedging this form, we obtain
+(9.15)
+ω ∧ ω
+=
+−γr−2βγ+2γ−1dr ∧ dv ∧ dθ ∧ dw
+=
+−γr
+2−3α
+2+α dr ∧ dv ∧ dθ ∧ dw.
+where we use (9.10). Let us set f(α) = 2−3α
+2+α . This function does not take values
+lower than −3 or higher than 1. When α = 2 this gives us a b-symplectic structure:
+ω ∧ ω = −γrdr ∧ dv ∧ dθ ∧ dw.
+The section of �4(bT M) given by the dual structure of ω ∧ ω is clearly transverse
+to the zero section.
+On the other hand if α = 2, then β = 1 and hence:
+ω = γr−1dr ∧ ω ∧ dv,
+and its dual Poisson structure is clearly also a proper section of �2(bT M).
+□
+Remark 9.3. One may ask if for other values of α it is possible to obtain
+other bm-symplectic structures for different m. For example for α = 6, as ω ∧ ω =
+−γr−2dr ∧dv ∧dθ ∧dw, so it seems likely to obtain a b2-symplectic form. But from
+the expression of ω it becomes clear that it is not a proper section of �2(b2T ∗M)
+
+9.4. POTENTIAL APPLICATIONS
+105
+m1 = 1 − µ
+m2 = µ
+q
+r2 = q − q2
+r1 = q − q1
+Center of mass
+r
+q1
+q2
+Figure 1. Scheme of the three-body problem.
+9.4. The restricted three-body problem
+In this last section of the monograph, we catch up with the circular planar
+restricted three-body problem.
+The restricted elliptic 3-body problem is a simplified version of the 3-body
+problem. It describes the trajectory of body with negligible mass moving in the
+gravitational field of two massive bodies called primaries, orbiting in elliptic Kep-
+lerian motion. The restricted planar version assumes that all motion occurs in a
+plane.
+The associated Hamiltonian of the particle can be written as:
+(9.16)
+H(q, p) = ∥p∥2
+2
++
+1 − µ
+∥q − q1∥ +
+µ
+∥q − q2∥ = T + U
+wit µ the reduced mass of the system.
+As it was observed in [KMS16b], it is possible to associate a singular struc-
+ture to this problem. Consider the symplectic form on T∗R2 in polar coordinates,
+After making a change to polar coordinates (q1, q2) = (r cos α, r sin α) and the
+corresponding canonical change of momenta we find the Hamiltonian function
+(9.17)
+H(r, α, Pr, Pα) = P 2
+r
+2 + P 2
+α
+2r2 + U(r cos α, r sin α)
+where Pr, Pα are the associated canonical momenta and with potential energy:
+U(r cos α, r sin α)
+The McGehee change of coordinates is used to examine the behavior of orbits
+near infinity, see also [DKdlRS19]:
+(9.18)
+r = 2
+x2 .
+The corresponding change for the canonical momenta is easily seen to be
+(9.19)
+Pr = −x3
+4 Px.
+
+106
+9.
+POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS
+The Hamiltonian is transformed to
+(9.20)
+H(r, α, Pr, Pα) = x6P 2
+x
+32
++ x4P 2
+α
+8
++ U(x, α).
+By transforming the position coordinate (9.18) without modifying the momentum
+associated to r, we are left with a simpler Hamiltonian, however, the pull-back of
+the symplectic form is no longer symplectic, but exhibits a singularity of order 3
+and it is called b3-symplectic:
+(9.21)
+ω = 4
+x3 dx ∧ dPr + dα ∧ dPα.
+Adding the line at infinity provides a description of the dynamics within the
+critical set Z = {x = 0}. From the change of coordinates implemented, we might
+think that the dynamics within Z may have no physical meaning, but its interplay
+with the dynamics close to Z gives information about the behaviour of escape orbits
+sometimes identified as singular periodic orbits (see [MO21] and [MOPS22]).
+Given an autonomous Hamiltonian system of a symplectic manifold of dimen-
+sion 2n, the level sets of the Hamiltonian function are often endowed with a contact
+structure ( a contact structure is given by a one form α satisfying a condition of
+type α ∧ (dα)n−1 ̸= 0).
+In [MO18, MO21] applications of the b-apparatus are discussed in this con-
+text. In particular, the notion of bm-contact structures is introduced by translating
+the condition above for bm-forms. The classical notions in the contact realm such
+as Reeb vector fields can also be introduced in this set-up.
+By considering the Mc Gehee change as we did in the contact context, in
+[MO21] it is proved:
+Theorem 9.4. After the McGehee change, the Liouville vector field Y = p ∂
+∂p is
+a b3-vector field that is everywhere transverse to the level sets of the Hamiltonian Σc
+for c > 0 and the level-sets (Σc, ιY ω) for c > 0 are b3-contact manifolds. Topologi-
+cally, the critical set of this contact manifold is a cylinder (which can be interpreted
+as a subset of the line at infinity) and the Reeb vector field admits infinitely many
+non-trivial periodic orbits on the critical set.
+One of the possible applications of our KAM theorem would be to find new
+periodic orbits of the restricted three body problem close to infinity by perturbing
+the periodic orbits described above. This old technique of perturbation theory is
+probably due to Poincar´e (Poincar´e’s continuation method, see [MO17]).
+This
+opens the door to new investigations which will be considered elsewhere.
+
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+New Series 42 (2010).
+[GMP14]
+Victor Guillemin, Eva Miranda, and Ana Rita Pires, Symplectic and poisson geom-
+etry on b-manifolds, Advances in Mathematics 264 (2014), 864–896.
+[GMPS15a] Victor Guillemin, Eva Miranda, Ana Rita Pires, and Geoffrey Scott, Toric actions
+on b-symplectic manifolds, Int. Math. Res. Not. IMRN (2015), no. 14, 5818–5848.
+MR 3384459
+[GMPS15b]
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+Res. Lett. 24 (2017), no. 2, 363–377. MR 3685275
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+Victor Guillemin, Eva Miranda, and Jonathan Weitsman, Desingularizing bm-
+symplectic structures, International Mathematics Research Notices 2019 (2017),
+no. 10, 2981–2998.
+[GMW18a]
+Victor W. Guillemin, Eva Miranda, and Jonathan Weitsman, Convexity of the mo-
+ment map image for torus actions on bm-symplectic manifolds, Philos. Trans. Roy.
+Soc. A 376 (2018), no. 2131, 20170420, 6. MR 3868423
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+(2018), 941–951. MR 3804693
+[GMW21]
+, On geometric quantization of bm-symplectic manifolds, Math. Z. 298 (2021),
+no. 1-2, 281–288. MR 4257086
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