Fabio Giuseppe Pusateri. ANALYTIC KAM TORI for the PLANETARY -BODY PROBLEM. Prof. Luigi Chierchia

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1 UNIVERSITÀ DELI STUDI ROMA TRE FACOLTÀ DI SCIENZE M.F.N. raduation Thesis in Mathematics of Fabio iuseppe Pusateri ANALYTIC KAM TORI for the PLANETARY -BODY PROBLEM Adviser Prof. Luigi Chierchia The Candidate The Adviser ACADEMIC YEAR October 2006 Key ords: Kolmogorov-Arnold-Moser theory small denominators Rüßmann s theory for analytic nearly integrable Hamiltonian systems properly degenerate Hamiltonian systems Arnold s Theorem on the stability of planetary motions.

2 Abstract In [Féj04] Jacques Féjoz completed and gave the details of Michel Herman s proof of Arnold s 1963 theorem on the stability of planetary motions. This result provided the existence of maximal invariant tori for the planetary -body problem ith in a neighborhood of Keplerian circular and coplanar movements under the hypothesis that the masses of the planets are sufficiently small ith respect to the mass of the Sun. In this thesis e prove an analogous result in analytic class i.e. e prove under the same hypotheses listed above the existence of real-analytic maximal invariant tori for the planetary -body problem. The proof is based on the cited article by J. Féjoz and on a 2001 paper by H. Rüßmann. First e prove a general quantitative theorem about existence of maximal KAM tori for nearly-integrable Hamiltonian systems near elliptic loer dimensional tori. Then using [Féj04] e obtain a set of initial data in the phase space of the Hamiltonian model for a planetary system ith strictly positive Lebesgue measure leading to quasi-periodic motions ith frequencies. In appendix A and B e give a complete and detailed proof of Kolmogorov s original 194 KAM theorem. In appendix C e briefly revie Rüßmann s theory contained in [Rüßm01] about loer dimensional elliptic invariant tori for nearlyintegrable Hamiltonian systems.

3 Contents 0.1 Notations Summary 7 2 Rüßmann s theory on maximal KAM tori Non-degeneracy conditions Remarks and examples Preliminaries to Rüßmann s theorem Statement of Rüßmann s Theorem on maximal KAM tori Estimate for the size of the perturbation in Rüßmann s theorem Existence of non-resonant frequencies Description of the iteration process Theorem about existence of non-resonant frequencies Theorems on the measure of a set defined by small divisors Links and Chains Properly degenerate Hamiltonian systems Statement of results Averaging theory Elliptic equilibrium for Birkhoff s normal form Symplectic polar coordinates Rüßmann s tori for properly degenerate systems Non-degeneration of the frequency application Index and amount of non-degeneracy Determination of and sufficient conditions to apply Rüßmann s theorem Choice of the initial compact set Parameters related to the initial compact set Choice of index and amount of non-degeneracy Norm of the frequency application

4 4.3. Approximation function and control on the small divisors Determination of Determination of Definition of Determination of The quantities Determination of and conditions on Conditions on in theorems and The spatial planetary -body problem Hamiltonian models for the planetary -body problem The secular Hamiltonian and its elliptic singularity Verification of Arnold-Pyartli condition Arnold-Pyartli condition in the plane Arnold-Pyartli condition in the space Proof of Arnold s theorem A Proof of Kolmogorov s 194 theorem on persistence of quasi-periodic motions 130 A.1 Introduction A.1.1 Some useful estimates A.1.2 Diffeomorphisms on A.2 Kolmogorov s idea and first step of the proof A.2.1 Reduction of the perturbation to order A.2.2 Control on the domain of A.2.3 Estimates on and A.3 Iteration and conclusion A.3.1 Inductive step and convergence of the scheme A.3.2 Final estimates A.4 Dependence on additional parameters B Measure of Kolmogorov s invariant tori 18 B.1 Introduction B.2 Kolmogorov s normal form for B.3 Extension of! B.4 Measure of invariant tori C Rüßmann s theory for loer dimensional elliptic tori 177 C.1 Preliminaries to Rüßmann s theorem C.1.1 Consequences of eakly non-degeneration C.1.2 Estimate for " #%$ &')

5 C.2 The main theorem C.2.1 Explicit estimate for in Rüßmann s theorem C.2.2 Brief explanation of the strategy adopted by Rüßmann C.2.3 Conditions to carry out the -th step of the iteration process191 C.2.4 Survey of the conditions to carry out the -th step of the iteration process for Bibliography

6 T 0.1 Notations 1. Let or alays means otherise specified. if not 2. Let and the usual -dimensional torus e define for and the folloing sets:! #" + -/. 10 $&%&') Im /61.7 Re /8 9;: <&<&<= 0> e ill also use sometimes the notation? instead of 2 ; if!7 e denote 1@ A BC < 3. Let ' a domain and3 ' EDF a real-analytic function e put H3!IKJ $ $&% H L and if can be holomorphically extended on a ' e denote MN HPOQRIKJ $ $&% HPOQ L S< 4. Let TUWVM ith VX or a domain and T EDYV [Z\V^]S_`] a a -times continuously differentiable function. We denote Ab% &<&<&< cb6d?v dfe EDhg d Ai Ab% &<&<&< cb6d 3V Z3V ]S_`] as the a -th derivative of in i +T ith the convention g d Ai Ab Ai for a ); moreover e rite g d Ai Ab d j g d Ai Ab &<&<&< cb and define g d Ai 1 kml # nc%porq/sutnvtyx g d Ai Ab d here zm is some norm in V {Z3V^]S_`]. For any 'T e rite g d Ai H3!IKJ $ % H g d Ai 1!~j< The Banach space of all -times continuously differentiable function ED Z ]S_`] here T is an open subset of ) ith bounded derivatives up to order is defined by AT K ƒz ]S_`] provided ith norm Ukml K /dv # g d.~ˆ<

7 . ) for O denotes the standard symplectic matrix ) < 6. For any : O e define 3 : 0 < Other definitions or notations that are used only in some parts of the thesis ill be introduced hen necessary. 6

8 Chapter 1 Summary...l Amor che move il sole e l altre stelle. Dante Alighieri. Divina Commedia Paradiso Canto XXXIII. The main body of this thesis is divided into four chapters and three appendices. In chapter 2 e revie H. Rüßmann s theory about existence of analytic maximal KAM tori for nearly integrable Hamiltonian systems. All results provided are obtained from the paper [Rüßm01] here actually Rüßmann proves a much more general result about existence of loer dimensional invariant tori. At the beginning of section 2.1 e introduce to definitions of eak nondegeneracy conditions; the term eak is used here to underline the difference ith respect to the classical non-degeneracy conditions in the history of KAM theory see for example A.N. Kolmogorov s 194 theorem in appendix A). The first definition presented is the so called Arnold-Pyartli condition used by M. Herman to prove a very general and elegant theorem about existence of invariant tori for nearly-integrable Hamiltonian systems. Immediately after the definitions of non-degeneracy eak non-degeneracy and extreme non-degeneracy in the sense of Rüßmann are given. Some simple examples and remarks follo together ith a brief discussion of the relations beteen the to different nondegeneracy conditions. In section 2.2 e prepare for the statement of Rüßmann s theorem on maximal invariant tori; e introduce the concept of approximation function used to control in a more general ay than the classical diophantine inequalities) the small denominators appearing in the problem and give the fundamental definition of index and amount of non-degeneracy of a real-analytic function. Soon after e present a somehat quantitative statement of Rüßmann s theorem obtained putting together different results contained in the article cited above. 7

9 In section 2.4 e provide an explicit estimate for the size of the perturbation in Rüßmann s theorem. We remark that this estimate has been deduced considering the case of maximal tori all along Rüßmann s paper and simplifying some numerical values ith the imposition of more strict upper bounds. As ending to this chapter in section 2. e discuss ith some more details Rüßmann s theorem about existence of non-resonant frequencies under the hypothesis of non-degeneration of the frequency application i.e. the gradient of the integrable part of the considered Hamiltonian function). We have chosen to investigate this aspect of Rüßmann s ork see [Rüßm01 Part IV] for complete details) not only for its technical beauty but to serve an important purpose. We aim to prove that it is not necessary to use the literal definition of index of nondegeneracy given by Rüßmann see section 2.2 and definition 2.2.2); indeed it ill be shon that this quantity may be replaced by any integer as ell as the related amount of non-degeneracy may be substituted by the amount corresponding to see remarks in subsection 4.3.3). Actually there is also another reason for the exposition of this matter: by the construction described the reader ill understand ho it is possible to control the non-resonant frequencies ith the classical diophantine inequalities instead of using an approximation function see subsection 4.3. for details). This aspect apparently trivial is fundamental to apply Rüßmann s theorem to properly degenerate Hamiltonian systems i.e. nearly-integrable systems hose integrable part does not depend upon all the action variables as for example the Hamiltonian model for the many-body problem. In chapter 3 e prove the first part of a general theorem about analytic and properly degenerate Hamiltonian systems. We infer existence of a positive measure set of initial data in the phase space of the considered Hamiltonian function leading to quasi-periodic motions laying on analytic Lagrangian maximal) KAM tori. This theorem is an analogous performed in analytic class of M. Herman s KAM theorem contained in [Her98] a proof can be also found in [Féj04]). Herman s theorem is extremely elegant and very general since it establishes also the existence of loer dimensional tori for properly degenerate Hamiltonian systems hile e ill only prove the existence of maximal KAM tori in the general theorem as ell as in the application to planetary systems). A non-properly degenerate version of Herman s theorem is Rüßmann s theorem discussed in chapter 2 case of maximal tori) and appendix C general case of loer dimensional tori). Indeed this last theorem is the one e ill apply to prove our KAM theorem. In section 3.1 e start introducing the general Hamiltonian setting. We consider a real-analytic nearly-integrable Hamiltonian in the form here is a small positive parameter; in particular e require that the average of the perturbation ith respect to the angles e shall call possesses an elliptic equi- 8

10 librium point say in the origin. Then e give general formulations of the results proved in chapter 3 and 4; theorem states the main result and is folloed by a brief description of the strategy of the proof made through theorems and The first of this three theorems is proved in the hole remaining of chapter 3. The scheme adopted is a classical one as ell as the majority of the results used and applied and reflects for most aspects Herman s scheme in his general KAM theorem see [Féj04 theorem 60]). The proof runs as follos. Section 3.2 contains the exposition of a ell knon theory of dynamical systems that is averaging theory. Using the hypothesis of non-degeneracy in the sense of Rüßmann of ; e localize the initial action variables in an open ball here e can apply a corollary of a general result taken from [Val03] theorem in this thesis). Averaging theory permits the removal of the dependence upon the angle variables up to any chosen and fixed order greater than. Unfortunately it causes at the same time a little shift of order of the elliptic equilibrium initially possessed by. Thus in section 3.3 e find a symplectic transformation that restores the original equilibrium. Subsequently e use a ell knon result by K.Weierstraß to diagonalize the quadratic part of the function obtained composing ith the symplectic transformations made so far. Next in section 3.4 a general formulation of a classical theorem usually called Birkhoff s normal form theorem is provided. Folloing the ideas in [Zeh94 pages 43-44] e give a detailed and quantitative proof about existence of a symplectic map that puts an Hamiltonian into Birkhoff s normal form up to a certain arbitrarily fixed order. The most important aspect to underline is that e give an explicit evaluation of the domains of the transformation. In the first part of section 3. e perform the passage to symplectic polar coordinates that represent angle-action variables for the integrable part of the Hamiltonian considered. Subsequently e make a rescaling of the ne angle-variables ith a conformally symplectic map hich casts the Hamiltonian in a simple form very similar to that considered in Rüßmann s theorem More precisely the Hamiltonian system obtained so far is a degenerate case of the system considered in Rüßmann s theorem i.e. the Hamiltonian function obtained has the same form of Rüßmann s Hamiltonian ith the difference that in the first the dependence upon the small parameter appears also in its integrable part. Chapter 4 is dedicated to the application of Rüßmann s theorem and constitutes the most original part of this thesis. In section 4.1 e analyze the frequency application of the integrable part of the Hamiltonian obtained through the different steps performed in chapter 3. This ne frequency application is related to the initial one by the folloing to aspects: their difference is and the last components of the ne frequency application that is the components given by the first Birkhoff s invariant) are 9

11 multiplied by a factor proportional to. Thus assuming sufficient condition on e prove that the frequency application of is non-degenerate in the sense of Rüßmann. Moreover imposing further upper bounds e are able to control the index and amount of non-degeneracy of this frequency application and relate them to the index and amount of the initial one. This key passage necessary to apply Rüßmann s theorem and at the same time to give an explicit estimate for the size of the initial perturbation is performed in section 4.2. The folloing section 4.3 describes ith full details the application of Rüßmann s theorem for maximal invariant tori. We consider and analyze every single quantity and constant involved in Rüßmann s estimate for the size of the perturbation; in particular e focus our attention on ho they change order in hen our degenerate case is taken into consideration. This allos us to determine in subsection a sufficient loer bound for the order to hich e may remove the dependence upon the initial angles ith averaging theorem) and the order up to hich e may apply Birkhoff s normal form theorem. Particular attention may be paid to subsection 4.3. here e explain the different choice ith respect to Rüßmann for the control of the small denominators and sho ho our choice is effectively possible in Rüßmann s theorem. In subsection e determine sufficient conditions on to apply Rüßmann s theorem and finally in section 4.4 e gather all other conditions imposed on in chapter 3 and in sections 4.1 and 4.2. The main part of this thesis ends ith chapter here e revie the results contained in [Féj04 pages 4-62]. In the cited pages J. Féjoz completes M. Herman s ork on the spatial planetary -body problem proving the nonplanarity of the planetary frequency application. Then applying Herman s general KAM theorem about invariant tori he finally obtains a proof of Arnold s 1963 theorem on planetary motions in the case theorem 63 in [Féj04] 1 ). In particular J. Féjoz proves that the spatial planetary frequency map formed by frequencies does not satisfy any linear relation on an open and dense subset ith full Lebesgue measure of the secular space i.e. the collisionless space of the ellipses described by the planets). This is equivalent to prove the non-degeneracy of in the sense of Rüßmann on the same open set and is the fundamental result e use to apply our KAM theorem for analytic invariant tori to the spatial planetary problem. 1 Actually in [Arn63b] V.I. Arnold announced a somehat stronger result: If the masses eccentricities and inclinations of the planets are sufficiently small then for the majority of initial conditions the true motion is conditionally periodic and differs little from Lagrangian motion ith suitable initial conditions throughout an infinite interval of time. Hoever Arnold proved this statement only in the case of the planar three-body problem and gave indications on ho to generalize this result but apparently nobody has ever succeeded in implementing Arnold s indications. 10

12 We can describe the structure of chapter as follos. First e introduce the general setting of the Hamiltonian model for the spatial planetary problem; starting from Neton s equation e pass to heliocentric coordinates and regard the problem considered as a perturbation of decoupled to-body problems. Further on in section.1 e perform the passage to Poincaré coordinates and describe their relations ith the elliptic orbital elements. In section.2 e define the secular Hamiltonian as the average made ith respect to the angle-variables of the perturbative function expressed in Poincaré variables. Then e summarize ho is it possible to put the secular Hamiltonian into the form considered in the general KAM theorem stated in section 3.1. In particular results contained in [Poi07] [Rob9] and [Las91] to hich e alays refer for all the details) are used to express ith the help of Laplace s coefficients the quadratic part of the secular Hamiltonian in a very remarkable and simple form. Section.3 starts ith some preliminary lemmata and a proposition concerning the eigenvalues of a symmetric matrix hich is a perturbation of a diagonal matrix. In subsection.3.1 e use the results just shon to check Arnold-Pyartli condition for the planetary -body problem in the plane. First of all the frequency map composed by frequencies is complexified on a certain connected domain using the analycity of the Hamiltonian function expressed in Poincaré variables and the holomorphic extension of Laplace s coefficients. Then the development of the quadratic part of the secular Hamiltonian for small ratios of the semi major axes yields the non-planarity of the frequency map on an open and dense subset of the secular space ith full Lebesgue measure. In subsection.3.2 the non-planarity of the frequency application for the spatial problem is discussed. It is immediately observed that the non-degeneracy condition is not verified in this case. In fact the spatial planetary frequency application satisfies to and only to) linear relations: the first is due to the presence of a null frequency hereas the second is a strange resonance remarked for the first time in his generality by M. Herman indeed it is usually called Herman s resonance ). To suppress this to linear relations Féjoz follos an intermediate strategy beteen Herman and Arnold. If e denote plt the Hamiltonian function of the spatial planetary problem expressed in Poincaré variables Féjoz idea is to consider a modified Hamiltonian plt here is a real parameter and denotes the third component of the total angular momentum. The frequency application of extends for all the frequency application of plt and does not verify Herman s resonance. Moreover the restriction to the symplectic submanifold of vertical total angular momentum suppresses also the linear relation given by the null eigenvalue so that the non-planarity of the frequency application is satisfied for. Thus theorem gives an open set of initial data ith full 11

13 measure leading to quasi periodic motions for the flo of laying on invariant analytic Lagrangian tori. Finally since plt and commute they possess the same Lagrangian tori so that the result is established also for plt. The second part of this thesis is constituted by three appendices: in appendix A and B e revie the classical KAM theory hile in appendix C a summary of the more recent Rüßmann s theory on the construction of analytic loer dimensional invariant tori is presented. In appendix A e discuss the result gave by Kolmogorov in his 194 theorem on the persistence of quasi-periodic motions for analytic nearly-integrable Hamiltonian systems. In [Kol4] the Russian mathematician stated a general theorem and outlined a brief sketch of the proof; he shoed ho periodic motions for integrable systems still persist under the addition of a perturbation provided that this last is small enough and assuming a non-degeneracy hypothesis on the quadratic part of the integrable part of the Hamiltonian function. Using his brilliant ideas e prove a quantitative version of his results theorem A.1.2) providing also an explicit estimate for the size of the perturbation. In subsections A.1.1 and A.1.2 some useful preliminary estimates are given. In section A.2 e describe Kolmogorov s main idea and perform the first step of the iteration process needed to put the Hamiltonian function considered into Kolmogorov s normal form. Then in section A.3 e discuss the iterative scheme and finally prove the convergence. Appendix B contains a classical result concerning the measure of maximal invariant tori carrying quasi-periodic motions usually called Kolmogorov s tori ) for nearly-integrable analytic Hamiltonian systems. A brief description of the general setting is made in the first to sections. In section B.3 e use Whitney s extension theorem formulated and proved in [Whi34]) to extend the symplectic map gave by Kolmogorov s theorem i.e. the map that puts the given Hamiltonian function into Kolmogorov s normal form) to a map. From the regularity of this extension ith the help of some classical diophantine estimates e obtain the main result in theorem B.4.1: the measure of the complement of the union of maximal invariant KAM tori in the phase space of a nearly-integrable analytic Hamiltonian system is proportional to the size of the perturbation. The thesis ends ith appendix C here e describe Rüßmann s theorem for the existence of loer dimensional tori in nearly-integrable Hamiltonian systems. This remarkable result contained in [Rüßm01] is obtained ith the hypothesis of eak and extreme non-degeneration in the sense of Rüßmann of the frequency application. We remark once again ho Rüßmann s theory for nearly-integrable Hamiltonian system and his concept of non-degeneracy are fundamental in our ork it is obvious!) but have also influenced Herman s ork on the matter and 12

14 vice versa 2. The contents of appendix C can be briefly described as follos. Section C.1 contains a bit of preliminaries: e recall the definition of approximation function; e expose ith some details the complete theory of index and amount of nondegeneracy for a eakly non-degenerate and extreme real-analytic function; e define the amount of degeneracy for such functions and give a useful estimate for the exponent of a matrix function. Then e state a quantitative version of Rüßmann s main theorem and in subsection C.2.1 e give an explicit estimate for the size of the perturbation. The to folloing subsections are dedicated to a brief description of Rüßmann s strategy and iterative scheme. Subsection C.2.4 contains the survey of Rüßmann s conditions needed to carry out the -step of the iterative scheme for. It is motivated by our statement of Rüßmann s theorem hich is slightly different from the original. In fact in our formulation e add a term to the Hamiltonian function considered by Rüßmann; then e verify that the initial conditions indicated by Rüßmann hold for this ne Hamiltonian so that e are alloed to enter Rüßmann s scheme ithout any further modifications. 2 In [Rüßm01] e read: I thank M. Herman for his permanent interest in the realization of this ork and I thank him also that he has emphasized at every turn that the discovery of invariant tori under the condition of eak non-degeneracy is my ork in particular in the case. here means the case of maximal tori). 13

15 Chapter 2 Rüßmann s theory on maximal KAM tori 2.1 Non-degeneracy conditions In this section e give definitions of some different kinds of non-degeneracy conditions under hich invariant tori for an unperturbed Hamiltonian system persist ith the addition of a small perturbation. In particular e are interested in the eak non-degeneracy conditions defined by A. S. Pyartli and idely used and developed by M.Herman and J.Féjoz in [Féj04]) and by H. Rüßmann. Definition Let EDU a parametrized curve of class here is a compact interval is said to be non-planar at if - - &<&<&< > 2.1.1) " is said to be non-planar homogeneously non-planar for Herman) if it is non- or equivalently if it is not contained in any vectorial planar at all points ƒ hyperplane). is said to be planar at f in ) if it is not non-planar at f in ). Instead is called essentially non-planar non-planar for Herman) if for every open subset ) the image ) is not contained in an affine hyperplane. Let T be a closed ball and T D a parametrized curve; is called non-planar at the point { T if there exist an immersion &D AT here is an interval of such that the curve is non-planar at & ; it is non-planar if it is non-planar at all point MT hile it is essentially non-planar if its local image is nohere contained in an affine hyperplane. The so called non-degeneracy condition of Arnold-Pyartli consists in the non-planarity of the frequency vector of the unperturbed Hamiltonian system. The - 14

16 ' most important issues related to this non-degeneracy condition and the properties of such non-degenerate functions are extensively discussed in [Pya69]. Definition Rüßmann non-degeneracy condition). Let T a domain a non-void open connected set) of : &<&<&< 4T ED is called nondegenerate if for any &<&<&< 0 a real-analytic function if and only if the range AT of does not lie in any -dimensional linear subspace of ); e call degenerate if it is not non-degenerate; a real-analytic function if is non-degenerate; defining the folloing subset of ' 3 c 3 c 6T ED Z ` c e call a eakly non-degenerate function 0 9 C3 is eakly non-degenerate c extreme if. 0 '! We remark that in this chapter e ill use exclusively the hypothesis of nondegeneration since e discuss Rüßmann s theory only in the case of maximal tori. On the other hand in appendix C the condition of eakly and extreme nondegeneracy ill be considered since Rüßmann s complete results in the case of loer dimensional tori are exposed Remarks and examples We no make some simple but important remarks: 1. We start remarking the equivalence of the non-planarity of a curve ED and the fact that is not contained in any vectorial hyperplane or any linear subspace of positive codimension in ). In fact if is contained in a linear subspace there exists b in such that b - for every ; by derivation b - b -!b - '. and therefore b - b - b - 1

17 : < for every m ; then since &<&<&< e have - - & " verify a linear combination - < 2.1.2) for every that is the definition of planarity. On the other hand if is planar on since it verifies equation 2.1.2) there exists i Ai &<&<&< ci 3 i such that i - i - - i for all 4. Let : { kml # &<&<&< M i 0 then by the last equation verifies on the ordinary differential equation of order : - i i - i - i - < 2.1.3) Choose no in the orthogonal space generated by the vectors - 0 x s s for some in ; then m - verifies equation 2.1.3) ith the initial conditions and therefore - - in It can be easily observed that the essential non-planarity implies the nonplanarity since is essentially non-planar from definition if for every open ) m ) 2.1.4) therefore by taking ) and using the continuity of e have - 9 m ) that is by the previous remark the non-planarity of 3. By remark 1 and definition e obtain for a curve %! ED the equivalence is non-planar in is non-degenerate in < 2.1.6) This equivalence underlines very ell the different aspect of the to definitions hich nevertheless describe the same geometrical property of a function. In particular the non-planarity describes a local behavior of the considered function hile Rüßmann s non-degeneracy condition has a global 16

18 nature. This is also clear from the to different settings in hich this definitions are given. The same equivalence in 2.1.6) holds if is defined on a closed ball T. 4. As far as an analytic function is considered the notion of essential nonplanarity can be expressed by - 9 m < 2.1.7) To be as much clearer as possible e provide some elementary examples of functions belonging to the different categories of non-degeneration: 1. A non-planar or non degenerate) but not essentially non-planar curve is - N ; in fact " - - for every hile - `. An example of a non-planar curve that is not essentially non-planar but hose global) image is not contained in any affine plane is D - s - s < 2. A degenerate function that is eakly non-degenerate but not extreme is c [ ED Z defined by - and - ; in fact ' hile `'! since - ` - ). 3. A degenerate function that is eakly non-degenerate and extreme is given by c since `'! '! ; an example of a degenerate function eakly non-degenerate and extreme ith `' ' is c. Next e state some simple results that ill help to better comprehend the strong relation beteen non-planarity and non-degeneration. As already observed the main difference consists in the regularity of the functions considered in this to definitions. In fact both may be regarded as geometrical properties ith the only difference that local properties of analytic functions may become global properties for instance the property of being not contained in any vectorial hyperplane) hile this is not true for finitely many times differentiable functions. Lemma Let a non-void interval of and a ED a parametrized curve of class. If the image of a is contained in a vectorial hyperplane of then function g - " a - a - &<&<&< a - is constantly vanishing on. On the other hand if g - for every there exists an open set ) such that a ) is contained in vectorial hyperplane. 17

19 Proof The proof of this result is quite immediate and can be found in [Féj04 page 33] Proposition Let 8T ED here T is a domain) a realanalytic function; if is non-planar at f+t then is non-degenerate in T. Proof By definition if is non-planar at in T there exists an immersion W D here is a compact interval) such that - and - is non-planar. Therefore denoting a - " a - a - &<&<&< a - by continuity there exists.. dist- such that " a - a - &<&<&< a - - e have for every ^ -. This implies by the preceding lemma that a is not contained in any vectorial hyperplane i.e. for all C7 it results that M a - is not constantly zero for 8. By the definition of a e have that for every 8 the function is never constantly vanishing for ; it follos that for every {- ^ does not constantly vanish on T hich the statement Another simple observation that ill turn out to be fundamental is the folloing: Remark Let T EDY a non-degenerate non-planar) function then +T Z 7 Z; DU defined by > for every m is non-degenerate non-planar) on TNZ. 2.2 Preliminaries to Rüßmann s theorem Definition Approximation function). A continuous function is called an approximation function if: 1. & - M for &.. ~ ; 2. so that & for any & ; 3. & & and 4. ED ; for any 7.~. so that IKJ $ & & S~ D. ~ for all! 18

20 In his ork [Rüßm01] Rüßmann uses an approximation function to perform in a most general ay the control on the small divisors; in fact since invariant tori have to be constructed requiring a priori certain arithmetical properties that the frequencies of quasi-periodic motions lying on this tori should have it is necessary to establish in advance a certain control on the small divisors appearing in the construction. This is done in the proof of Kolmogorov s theorem see appendix A) by requiring for the frequency vector the diophantine condition. Hoever it ould be clearer in the statement of theorem the role played by the approximation function. As a possible choice of an approximation function e indicate L 3 $ for a chosen. 1. Lemma Letp81T EDU be real-analytic and non-degenerate functions defined on a domain TN7 for each : <&<&<. Consider Z-)Z; and let &<&<&< ^ be some parameters. If e define ZT EDh as the real-analytic function ith respect to the variables) and as the folloing subset of x v 2.2.1) &<&<&< 9;: <&<&< 0 then for any non-void compact set T there exist numbers and such that kml # K g 2.2.2) here the derivatives are obviously taken ith respect to the variables). Proof The proof of this lemma can be found in [Rüßm01 page 18] Observe that if e consider a function Z+T ED kml # K /dv as in 2.2.1) then the function g d 1 We remark in advance that hen e ill try to apply Rüßmann s theorem for our purposes e ill not be able to use an approximation function to control the small denominators but e ill be forced to use classical diophantine inequalities. Details can be found in section

21 is continuous in Z3T for every. Therefore the number = k S%Es t t x kml # K /dv g d is ell defined for any compact set T and verifies = = for every N {N. Then by Lemma e can ell define the numbers and = as follos Definition We call index of non-degeneracy of ith respect to the first integer such that = M hile = for every.7 ); e call the number = amount of non-degeneracy of ith respect to. No e give the definition of the index and amount of non-degeneracy of the real-analytic function. In Rüßmann s theorem such function ill be the frequency map of the unperturbed Hamiltonian system considered. Definition Let N be a compact set T N a domain containing and T ED a real-analytic and non-degenerate function. Let X! 6 0 e define O the index of nondegeneracy of ith respect to as the first integer such that here - k S%Es % kml # K /dv g d 2.2.3) is called amount of non-degeneracy of ith respect to. We just observe that this definition is ell posed since in the form considered in 2.2.1) ith and is a function ith the only difference that the parameters are not independently varying in!z!7 Z but have been chosen to coincide. 2.3 Statement of Rüßmann s Theorem on maximal KAM tori As far as e consider only the case of maximal tori in Rüßmann s notation) the main theorem contained in [Rüßm01] can be formulated as follos: Theorem Rüßmann s theorem for maximal tori). Let be an open connected set of and L 2.3.1) L 20

22 a real-analytic Hamiltonian defined for L Z endoed ith the standard symplectic form. Let g a complex domain on hich can be holomorphically extended; let be any non-empty compact subset of ith positive -dimensional Lebesgue measure meas and let g be an open set such that < 2.3.2) Choose + such that Z MZ 2.3.3) Define O 2.3.4) and assume that the function ED is non-degenerate. Then for any ith. ;. meas there exist positive numbers and that assuming see section 2.4 for details) depending on C - such 2.3.) and taking real numbers verifying 2.3.6) 2.3.7) there exists a compact subset ith meas meas and a bi-lipschitz mapping Ai Z Z EDFg 2.3.8) here is an open neighborhood of the origin in such that the mapping e ED L Ai 2.3.9) defines for every i! an holomorphic canonical transformation on 2 z Z and for sufficiently small m Z ; P "! Z! 2 Z 21

23 the transformed Hamiltonian is in the form: Ai for every i and Ai 2 Z ; Ai c the ne frequency vector satisfies for all i in the diophantine inequality Ai c ) Finally e observe that the transformed Hamiltonian system possesses the solutions so that the system described by L Ai 0 < in 2.3.1) possesses the invariant torus Ai for in ith quasi-periodic flo ) for all i in ) 2.4 Estimate for the size of the perturbation in Rüßmann s theorem We no display an admissible value of the size of the perturbation) in remarking that the result e give here is obtained by Rüßmann s estimate in [Rüßm01 page 171] considering the case of maximal tori ) all along the procedure folloed to reach such an estimate; moreover the presence of some numerical quantities just absolute numbers) has been simplified by imposing more strict upper bounds. To be as much clearer as possible e recall briefly the role played by the different quantities involved in the estimate alays referring to C.2.1 for definitions and further explanations: is the amount of non-degeneracy of the real-analytic and nondegenerate frequency map hile is its index of nondegeneracy; observe that for a non-degenerate function the amount of degeneracy equals. 22

24 and are chosen as in the statement and is defined as the diameter of i.e. IKJ $ $&s S% ; let S~ EDh the chosen approximation function in according ith definition e choose and fix such that - O and the folloing inequality holds - r > 2.4.1) moreover e define p IKJ $ & &. ~ that is bounded by hypothesis on ); We define O ) and O and set q O? ` O O p! > 2.4.3) 2.4.4) k ) O ) Let ED defined by L 1. e consider & & - P 23

25 and then define for ˆ ith IKJ $ nc% qes tnvt x ; x : g & Ab & 2.4.7) here then a possible value for is given by U - O - Lk O O 0 k K = 2.4.8) < 2.4.9) 2. Existence of non-resonant frequencies We no follo Rüßmann s ork to sho ho maximal invariant tori for the nearly-integrable Hamiltonian system considered in can be constructed. We alays refer to [Rüßm01 pages ] for further details and for the proofs of results e just cite here as ell as for the complete result concerning the construction of loer dimensional invariant tori Description of the iteration process The initial condition of the iteration process are: 1. EDh is a real-analytic function defined on a domain an open and connected set) 7 that is non-degenerate in the sense of Rüßmann see definition 2.1.2); 2. is a chosen and fixed compact set ith meas ; 24

26 3. + is chosen small enough to + and is the holomorphic extension of ; it follos that is bounded on 4. f f ith so that e can ell define N O < 2..1) We ill often use the same notation to identify both the real function and its holomorphic extension hen there ill be no ambiguity. to be determined and f.. and At the beginning of the iteration process e take.;.+. f k 0 d d d d for a &<&<&< 2..2) here positive numbers + ill be later determined. Moreover is a chosen approximation function see definition 2.2.1) and - O - for any ; is a fixed real number and for a. a M. a ). Ed - observe that by the properties of d 2..3) e obtain.ed.ed for any With the frameork described the general step of the iteration process orks out recursively as e shall see no; if d d d d and d d are given for some a here and have been set before) then e determine dco dco and dco { AT dco as follos: e choose an arbitrary function d AT d such that d/ 8 d 2..4) and define d i d d Ai then e take dco d and if d dco dco dco -Ed 9 e set 2 dco d. Ed > 2..) d/ < 2..6)

27 2..2 Theorem about existence of non-resonant frequencies The folloing theorem [Rüßm01 theorem 16.7]) states the possibility to carry out the general step of the iteration process infinitely many times assuring the existence of non-resonant frequency vectors at each step as ell as the convergence of the scheme. Theorem 2..1 Existence of non-resonant frequency vectors). For any meas 2..7) there exist positive real numbers + such that choosing f 2..8) the iteration process described above can be carried out for any a ith the resulting sets d and the functions d possessing the folloing properties: 1. ith d > 2..9) d x 2. the sequence d 0 d % converges uniformly on that is 3. let therefore continuous on then i meas Furthermore let f O and - ); k S% kml # K /dv Ai c 9 C3 to a function ) meas meas < 2..11) be such that g d 2..12) for any 3 ith 6 as defined in 2..1) diam and 26

28 as defined in 2.4.7); then e can take ith O = accordingly to section 2.4. q - O 1 O? ` 1! ) 2..14) 2..1) 2..16) 2..17) O O p? 2..18) We dedicate the remaining part of this section to the explanation of the proof that Rüßmann gives to this theorem. We ill focus our attention on some parts and underline the fact that e are searching for the existence of maximal tori performing some proofs of intermediate results in the case ith refer to Rüßmann s notation) Theorems on the measure of a set defined by small divisors We start citing an important result given and proved by Rüßmann in [Rüßm01 pages ]: Theorem Let X be a non-empty compact set ith diameter IKJ $ $&s S% 1 and define /@ for some +. Let O K for some f ) be a function such that k S% K /dv kml # g d 2..19) ith. Then for any function K satisfying ˆ and any ) r 27

29 g e can estimate meas ith F 6 0 T{ T q?? ` O! O 2..21) < 2..22) A proof of this theorem in the case can be found in [Pya69] here it appeared for the first time. Here e state a simple lemma that e are going to use in next theorem s proof: Lemma Let then it results here P is defined in K M_r for some open set and C [ z C Proof The proof is immediate and can be easily obtained by Leibniz rule g d Ai Ab d O6 x d for any bm ith b and i g d Ai Ab d a 1 O6 x d : g a : Ai Ab Ai Ab. Therefore g g Ai Ab Ai Ab z C hich gives the statement if e take the IKJ $ over b i and a. Let ED a real-analytic and non-degenerate function let be a compact set ith diameter IKJ $ $&s S% f and take such We recall that by lemma C.1.2 there exist numbers f and - such that k S% K /dv kml # g d 2..23) for every. Then e can state the folloing result: Theorem 2..3 Measure of a set defined by small divisors). Let as considered above and T K satisfying the estimates O 2..24) O )

30 < > for some and let the chosen approximation function see definition 2.2.1). Moreover let.. meas and. + O q ` ith then the measure of the set can be estimated by Furthermore if i h4 O?? meas O p! Ai c 9 C ) > 2..27) ) meas < 2..29) e can take We remark that a complete proof of this result even in the case and ED ith can be found in [Rüßm01 pages ]. Proof Let and C by hypotheses e have 4 T K. Setting accordingly to C.1.7) in hich e consider also the case of frequencies c of loer dimensional tori) it results No for every a i and bm ith b e have g d Ai Ab d cg d Ai Ab d g d Ai Ab d 1 having used the definition of v for and. in Since analogous estimate can be done instead of taking the IKJ $ over b ci and a e obtain 29

31 observe that for hat concerns the estimate regarding only e can replace ith any integer identity for instance) since is real-analytic. We no rite the obtaining by the preceding estimates the definition of 2..1 for the -norm of a product m< hich imply respec- O Furthermore if e consider the to cases tively and e have 4 O O and O!! O! No e are in a position to apply theorem 2..2 ith O K 4 K and henever and ith given by q?? meas i O T!T obtaining. r ` 30! and and lemma! O < [ uo O K 2..30) 2..31) 2..32)

32 . <.. e < see 2..21) 2..22) and the estimate on in 2..28) e have i h4 1i h4 i F Mi h4 Then in vie of 2..30) applied ith it results meas O ). No by the definition of Ai 6. c 9 C3 Ai. - c. - c c c meas " i F % q meas ^i F %1q %1q c 9 C3 9 C3 9 C3 To estimate this last sum e observe that the function &; is strictly decreasing because 2.2.1); therefore in vie of c x x c c 0 0 < 2..33) S~ D & & is itself decreasing see condition in definition c for any C3 e have d x t t x d a a Denoting no dƒ t t /d a for a O and e 31

33 d < can continue the above chain of equalities riting d x d x d x t t x d d a a a a a & & d d x dco & & & & d x & d d a a & & & & & & & & & a a & & having used in this last equality the property & for and recall once again definition 2.2.1). No since & & e may rite E E & p &1 p & & &1 O D & & &! & p & & & & 2..34) Finally e come back to the beginning of this chains of inequalities estimating meas meas meas O p < 2..3) Therefore in order to get 2..29) that is meas require O p 32 meas e must

34 as in 2..26) together ith 2..27) and in vie of as taken in 2..32). To complete the proof e still have to assure that condition 2..31) is satisfied ith our choice of in 2..33). From the property verified by in 2..23) hich is a consequence of its non-degeneracy as e sho in appendix C) e have - O O hich is satisfied by means of - \ 2..36) having used the estimate done before during the proof. From this to inequalities the definition of in 2..27) and p e derive O? O p? O? and by hypothesis e have No observe that the first terms that constitute O q? < 2..37). meas ` < 2..38) `! in 2..26) verify ` so that e obtain together ith 2..33) 2..26) 2..37) and 2..38) - ` 1 O? ` To complete the proof e observe that if e obtain / r r from 2..2) and hich gives < 2..39) Thus taking m ith e have that also 4 is a common upper bound for and as shon. Continuing this procedure e obtain a sequence of common upper bounds ; KO moreover from the recursive relation e can infer that the sequence tends to so that e can effectively take as a common upper bounds for the to norms considered 33

35 2..4 Links and Chains In the folloing pages e complete the proof of theorem 2..1 displaying Rüßmann s theory of links and chain for the construction of non-resonant frequencies on an arbitrarily non-empty compact set contained in the domain of the actionvariables. The full detailed construction of non-resonant frequencies in the case of loer dimensional tori) can be found in [Rüßm01 pages ]. Definition Let d!u a non-empty set and d! d d d see 2..2) for d ) e call d d d a link. The initial link in our iterative scheme is given by as in subsection Definition A linkmd recall d i d d d Ai d is said to be open if d -Ed 9 see subsection 2..1 for the definition ofd ). d K here ith and here e. Ed 2..40) Definition If a linkmd is open e call a linkmdco dco successor ofmd if here dco d dco d d d is an arbitrarily chosen function in d/ [ according to the frameork described in subsection 2..1). d dco a dco dco dco 2..41) d K satisfying the estimate 2..42) Finally e denote a if ; a. ~ or. a ~ and give the folloing Definition A collection of links K d &<&<&< d if a.~ &<&<&< if a ~ is called a chain if O is a successor of for any. a. The initial link is itself considered as a chain. A chain is said to be maximal if a. ~ and d is not open i.e. d ) or a ~ 34

36 ) e give here the folloing result: Since e ould like to ork ith -functions same open set say &<&<&< defined on the Theorem Let be a non-empty set and ED be an holomorphic and bounded function assuming real values on e ill alays think ). Then there exists a -function EDR such that L L for every and the folloing estimates hold for every &<&<&< : IKJ $ $&% q IKJ $ nc% qs tnvt x g d L Ab d a d IKJ $ $&% L 2..43) O ith a not depending on and increasing in a as defined in 2.4.7). Let the initial link e define its extension by f ith 1@. Thus e may attach to the chain the extended here chain. We ant no to define the extended chain of any assigned chain. Consider K for 3. a i.e. the functions defining the chain e may apply to each of this functions theorem 2..4 ith &Q such that obtaining for Moreover by the estimate in 2..43) e have g 1 = 1 ;. a < 2..44) in vie of 2..42). No e are in position to define recursively -functions on 1@ as follos f O 2..4) for any. a. Furthermore by induction e can easily obtain for a < 2..46) As this recursively construction shos each link of the chain K d possesses a ell defined extension. Therefore in the sense described e can call O a successor of accordingly to O O O 3

37 > definitions and define the extended chain K d as an extension of the chain K d. &<&<&< d if a.~ &<&<&< if a ~ Lemma 2..2 Estimates for extended chains 1). Let ˆ. here is chosen accordingly to 2..2)) and K d be a chain ith extension K d. Then for every g and consequently 2 [ a a it holds ` = ) ) 1 here = is defined in 2.4.7). Moreover for any a it results = 1 Proof From the recursive construction described before e get and ith estimate 2..44) e have 2 g = 2 [ x 2 As a consequence e get for every : g no since 8. and g x 2 1 = C inequality is increasing and therefore g ˆkml K `c # = &<&<&< =. : x < 1 : is increasing in : the right member of the above [ = 36 1 < 2

38 d From this last inequality and Cauchy s estimate e infer ` ` = Lemma 2..3 Estimates for extended chains 2). Let and extended chain ith and assume that in 2..2) satisfies f - E< 2..49) Then e have for any a ; furthermore for any function estimates and for any ; a e have d here K /d 1 [ 2..0) K verifying the C 2..1) C d i h4 %1qcs t t Ai c ) 2..3) Proof To prove 2..0) it is sufficient to observe that from 2..4) ith e have [ - having used hypothesis in 2..49) + and -. For hat concerns the remaining part of the statement e just have to prove x d &<&<&< d and the definition of d in 2..40). We start shoing since the other inclusions in 2..3) are given by the construction of the compacts i E Ai )