KAM theory:thelegacy of Kolmogorov’s 1954paper

HenkW. BroerUniversityof Groningen

Departmentof MathematicsandComputingScienceBlauwborgje3

NL-9747AC, Groningen,Email: broer@math.rug.nl

December30,2003

Abstract

Kolmogorov-Arnol’d-Moser(or KAM) theorywasdevelopedfor conserva-tivedynamicalsystemsthatarenearlyintegrable.Integrablesystemsin theirphasespaceusually contain lots of invariant tori and KAM theory estab-lishespersistenceresultsfor suchtori, which carryquasi-periodicmotions.We sketch this theory which begins with Kolmogorov’s pioneeringwork[40, 41].

1 Introduction

At theInternationalMathematicalCongressof 1954,heldatAmsterdam,A.N. Kol-mogorov gave a closinglecturewith title “The generaltheoryof dynamicalsys-temsandclassicalmechanics”[41], discussingthe paper[40]. The event tookplacein theAmsterdamConcertgebouwandit hasplayeda majorrole in thede-velopmentsof whatis now calledKolmogorov-Arnold-Moser(or KAM) theory.

In this lectureKolmogorov discussesthe occurrenceof multi- or quasi-periodicmotions,which in thephasespaceareconfinedto invarianttori. He restrictshim-self to conservative (or Hamiltonian)dynamicalsystems,asthesearegenerallyusedfor modellingin classicalmechanics.Invariant(Lagrangean)tori thatcarryquasi-periodicmotionswerewell-known to occurin Liouville integrablesystems

1

andKolmogorov’slecturedealswith theirpersistenceundersmall,non-integrableperturbationof theHamiltonian.The generalmessageof KAM theory is that generically, in small perturbationsof integrablesystemsthe union of quasi-periodicLagrangeantori haspositive(Liouville) measurebothin thephasespaceandin theenergy hypersurfaces.

On theonehandthis statementappliesto many concretemodelsof classicalme-chanics.Examplesarecertainversionsof the � -bodyproblem,of which thegen-eral non-integrability alreadyhadbeenestablishedby Poincare. Historically itsconsequencesfor philosophicalissueslike thestability of thesolarsystemareofinterest.Indeed,perpetualstabilityof thesolarsystemcertainlywouldbegrantedif its actualmotionwerequasi-periodic.On theotherhand,andfrom a moreglobalpoint of view, themeasure-theoreticalpartof KAM theoryimpliesthatfor typical Hamiltoniansystemsin finitely manydegreesof freedom,no ergodicity holds,sincethe energy hypersurfacescanbedecomposedin severaldisjoint invariantsetsof positive measure.This is of par-ticular interestfor statisticalphysics,wheretheergodichypothesisroughlyclaimsthatthesystem,whenconfinedto boundedenergy hypersurfaces,is ergodic.This paradoxprobablyis resolvedasthenumberof particlesis increasingsincethe obstructionto ergodicity provided by the KAM tori then seemsto decreaserapidly in importance.

Thepresentpapersketchescertaindevelopmentsrelatedto theKolmogorov 1954lecture[41]. We startwith discussingthelinearizationproblemnearafixedpointof ananalyticmapof thecomplex plane.This is thefirst known occasionwhereasmalldivisorproblemwassolved,thekind of problemwhich is socentralin KAM

theory. Next we turn to thedynamicsof circle maps,which goesbackto Arnold,followedby adigressionto areapreservingtwist maps,wherethenameof Mosercomesin. After this we discussconservative KAM theory in greatergenerality,alsotouchingthe ergodic ‘paradox’. For an earlierversionof the presentpapersee[12].

Remark A propertyis calledtypical if it occurson anopensetof thedynamicalsystemsat hand. Regardingthe topologyon ‘function space’,onemay think ofa (weak) Whitney topology for differentiablesystems,or of the compact-opentopologyfor analyticsystems.Thepropertywe have in mind is theoccurrenceinphasespaceof quasi-periodicinvarianttori with positivemeasure.Comparewith,e.g.,[18, 23].

2

2 Complex linearization

Considera local holomorphicmap(or a germ)��������� ��� �������

of the form����� �����������������with

���� �����! "�� �#�$&%Thequestionis whetherthereexistsa

local biholomorphictransformation' �!����� �(� ����� �suchthat'*) �+�$�, ' % (1)

We saythat ' linearizes�

nearits fixedpoint.

Formal solution

First considerthe problemat a formal level. The correspondingresultsgo backto Poincare, for an overview seeArnold [6]. Given a seriesexpansion

����� �-�.0/�132 � / � /welook for anotherseries' ��� �#�4�#� .0/�132!5 / � / � suchthattheconju-

gationrelation(1) holdsformally. It turnsout thata formal solutionexistswhen-ever

�76� is not a root of 1. Indeed,the coefficients

5 /can be determined

recursively by thefollowing equations:�8�:9<;=�!� 5 2 � � 2 ��8�:9<;>� 2 � 5�? � � ? ��@��!� 2 5 2 �(2)�8�:9<;>�BADCFEG� 5 A � � A � alreadyknown terms

%Fromthis theclaimdirectly follows.

Convergence

In thehyperboliccasewhereIHKJL��J(6�M9

theseriesfor ' haspositive radiusofconvergence.Thiswasprovenby Poincare,notby consideringtheseriesbut by adirectiterationmethod,compare[6].So thereremainsthe elliptic casewith

�ONQP E(the complex unit circle). We

observefrom theequations(2) that,evenif�

is notarootof unity, its powersmayaccumulateon

9�%This would givesmalldivisorsin theformal seriesof ' � which

castsdoubton its convergence.Thisproblemwassuccesfullysolvedby C.L. Siegel [66] in 1942.To thispurpose,writing

�$�SR 2UTWVLX �the following Diophantineconditionswere introduced. For

someY[Z and \]Z it is requiredthatJ_^I;a` bcJed Ybgfh�for all rationals!i b (with

b Z ). It turnsout thatthis is sufficientfor convergenceof theformalsolutionfor ' %

3

Let us discusscertainaspectsof the set of�jNOP E

satisfyingthe Diophantineconditions,as theseare relevant for us. It canbe directly shown that for fixedY=Z and \kZ @3� with Y sufficiently small, theDiophantineconditionsgive riseto a Cantorsubset

��N+P Ewhich is of almostfull measure.We recall that the

characterizing(topological)propertiesof a Cantorsetarecompactness,nowheredensenessandperfectness.It canbeeasilyseenthat themeasureof thecomple-ment is of order l � Y � as Y�m n% Comparewith, e.g., [18, 23]. We notethat forsmallvaluesof Y (aswell asfor largevaluesof \ ) only a small radiusof conver-gencefor ' canbeguaranteed[66].

Geometry and number theory

That the Diophantineconditionsgive rise to a nowheredensesubsetofP E �

di-rectly follows from the fact that the rootsof unity, which areeverywheredense,arein their complement.Thefollowing example,dueto H. Cremer[25] in 1927,alreadyindicatesthatthereis moreto this. For a nicedescriptionof this examplein asomewhatdifferentcontext see[11].

Considerthemap�o�n�p� �

givenby������#�$�!����� 2 �where

�[NIP Eis notarootof unity. Weshallseethatthereis a topologicallylarge

subsetof�

for which themaphasperiodicpointsin any neighbourhoodofn%

Forsuch

�this providesa geometricalproof that theformal conjugationdiverges,as

follows. Indeed,sincethelinearmap�rq� �!�

is anirrationalrotationall of whoseorbits that areeverywheredensein

P E �the existenceof periodicorbits in every

neighbourhoodof

impliesthattheformal linearizationmusthavezeroradiusofconvergence.

To examineperiodicpointsof periodb �

weconsidertheequation�tsu��� ���$�&%(3)

Usingthat �vsw��� �����Fsx�y�z,w,w,w�>� 2U{it follows that �tsW�����(;=�|�$�F����s(;}9c�0,w,w,w��� 2 { CFE �~%Abbreviating � ��@ s ;}9��

let� E ,�� 2 �u,w,w,����w� bethenontrivial solutionsof (3). It

follows thattheir productsatifiestheequation� E ,u� 2 ,#,w,w,(,g�w�=�$�Fs(;}9�%4

Fromthis weseethatthereexistsasolutionwithin radiusJ_� s ;�9nJ�E�� �of�r�z&%

Now considerthesetof�]N�P��

satifying���������&� sU��� JL� s ;�9&J�E�� � �$n%This setturnsout to be residual.1 We recall thata residualsetcontainsa count-able intersectionof dense-opensets;this propertymeansthat the set is large inthe senseof topology. The presentresidualset resemblesthat of the Liouville-numbers.Noticethatour residualsethasmeasurezero,sinceit hasall Diophan-tinenumbersin its complement.For ageneralbackgroundreferencedealingwithsubsetsof � that arelarge in topologyandsmall in measure,or vice versa,seeOxtoby[57].

Weconcludethatfor all�

containedin thisresidualset,periodicpointsof�

occurin any neighbourhoodof

�0� n�which makes the above divergenceargument

work.

By the endof the 20th centuryJ.-C.Yoccoz[69, 70] completelysolved the el-liptic case,proving that theso-calledBruno-condition[21] on thecorrespondingcontinuedfractions,is necessaryandsufficient for convergenceof theformal lin-earizationfor all possiblehigherorderterms.

3 Circle Maps

Weslightly shift context by consideringasmooth(for definiteness,say � � or realanalytic)family of circlediffeomorphisms� X�� � �eP�E(� P�E~�B��q������@���^��= D¡!�"���x^c�¢ ��~�where

 is asmallparameter;sowe arein theneighbourhoodof a family of rigid

circle rotations.For conveniencewe rewrite this asa ‘vertical’ family of cylindermaps� � �eP�E¤£¦¥L&�u9¨§!� P�E©£=¥L&�u9¨§ª�8�«����^��(q� �"���a@���^-�= D¡!�"����^#�¢ ��~��^¬�~%Theproblemis to find a diffeomorphism' � thatconjugates

� �and

�®­W%Notethat

the latter map is associatedto the family of rigid rotations. To be precise,werequirethatthefollowing diagramcommutes:P E £=¥L&�u9W§ ¯w°;h� P E £=¥L&�u9W§± ' � ± ' �P E £=¥L&�u9W§ ¯w²;h� P E £¦¥L&�u9W§��

1In otherwordsa dense³(´ or asetof 2ndBairecategory.

5

meaningthat � � )y' � � ' � ) �8­w% (4)

Small divisors again

We proceedby formally solving equation(4). Indeed,assumingthat ' � hastheform ' � �"���x^����j�«���=  µr�"���x^c�¢ ��~�x^¶�= �·���^c�¢ ��¢�

(5)

it follows that(4) canberewrittenasµ��«�y�¸@���^c�x^c�¢ ��!;�µr�"����^#�¢ ����$@g�¹·���^#�¢ ��&��¡����r�=  µr�"����^#�¢ ��~��^-�= �·���^#�¢ ��~�¢ ��Expandingin powersof

 andcomparinglowestordercoefficients,equation(4)

leadsto thelinearequationµ®­��«����@���^c�x^��®;�µ®­��«����^����$@g�¹·F­���^¬����¡ ­��"���x^��~%This linearequationcanbedirectly solvedby Fourier-series.Indeed,writing¡ ­��"����^¬�º� » ¼¨½u¾¤¡ ­ ¼ ��^¬�:R V ¼¢¿ �

µ®­��"����^¬�º� » ¼¨½u¾ µ®­ ¼ ��^¬�:R V ¼¢¿ �wefind that

·F­��p; E2UT ¡ ­�­w�which amountsto aparametershift, andthatµ®­ ¼ ��^¬�(� ¡ ­ ¼ ��^¬�R 2UT¨V ¼ X ;}9 %

Similar to what we hadin the previous case,we observe that thereexists a for-mal solutionassoonas

^is irrational. As before,the powersof

R 2UTWVLXmay still

accumulateon9�%

This givessmall divisors in the Fourier series,which puts itsconvergencein doubt.

A KAM theorem for circle maps

To overcomethe problemof small divisors,we introducethe sameDiophantineconditionsasbefore,requiringthat for givennumbers\*Z @

and Y=Z &� andforall rationals!i b onehas J_^I;a` bcJed Yb f %

6

In correspondenceto whatwassaidearlier, this givesaCantorset¥L&�u9¨§�ÀcÁj¥L&�u9¨§ª�

of almostfull measureas Y¦Z is small. As our first exampleof anactualKAM

theoremwenow formulate

Theorem 1 In the above circumstancesassumethat YzZ is sufficiently small.

Then,forJ  !J

sufficiently small, thereexistsa smooth( � � ) transformationof thecylinder ' � �3P E £=¥L&�u9W§Â� P E £=¥Ãn�u9W§��

conjugatingtherestriction�®­�J Ä�Å:ÆnÇ ­ � E«ÈLÉ to

a subsystemof� � %

Moreover, ' dependssmoothlyon 3%

This resultgoesbackto V.I. Arnold [5], alsocomparewith [6]. Thepresentfor-mulationcloselyfits with [19, 18]. Concerningthe smoothnessof the map ' � �comparewith Poschel[58] andwith [67, 71, 72]. The actualproof of theorem1 doesnot dealwith thepower seriesin

 3%Insteadit usesa Newton methodand

anapproximationpropertyby analyticmaps,of Whitney smoothmapsdefinedonclosedsets.

We observe thattheKAM theorem1 hasa stronglyperturbativecharacter:indeedit only appliesto small perturbationsof the rigid rotationfamily. In contrasttothis M.R. Herman[35] andJ.-C.Yoccoz[68, 69] have provena non-perturbativeversionof this theorem,i.e., for largevaluesof

 n%Also see[49].

Discussion

A circle diffeomorphismsmoothlyconjugatedto a rigid rotation�aq� �¶�$@���^

with^

irrational,is calledquasi-periodic. It is nothardto show thateachorbit ofsucha quasi-periodicmapfills thecircle densely[26]. Noticethatfor

^aNa¥L&�u9W§�Àthemaprigid rotation

� X�� ­certainlyis quasi-periodic.

A first consequenceof theorem1 thenis that thecircle maps� X�� �

thatareconju-gatedto oneof theDiophantine– andhencequasi-periodic– rotations

� X�� ­u�are

still quasi-periodic.So we concludethat quasi-periodicitytypically occurswithpositive measurein the parameterspace.Moreover, the fact that a Cantorsetisperfect,2 implies that in this settingquasi-periodicitynever occursasan isolatedphenomenon.

As aconcreteexampleconsidertheArnold family� X�� � �"�Â���z�r�a@���^��¦ �Ê ��� �of circle maps,looking at the

��^c�¢ ��-planeof parameters.We restrictto

J  �JhHË9��to ensurethat

� �is a diffeomorphism.It is known [5, 6, 26] that from thepoints��^c�G ����M� `�i b �x � resonancetonguesemanateinto theopenhalf-planes

 Ì6�on�in

sucha way that for smallJ  !J

anopenanddensesubsetis covered. In the tongue

2Whichmeansthatit containsno isolatedpoints.

7

emanatingfrom��^c�¢ ��#�Ë� `!i b �� � thedynamicsis periodic,with rotationnumber`!i b (for adefinitionsee,e.g.,[26]).

Theorem1 implies that in the complementof this union of tonguesthereexistsa unionof smoothcurvesthatfill out positive measure.For parametervaluesonthesecurves,thedynamicsis quasi-periodic.

Theorem1 hasanumberof applications.Indeed,it applieswheneverasystemofordinarydifferentialequationshasa 2-torusattractor, of which

� �is a Poincare

mapcloseto a rigid rotationfamily. In thatcasethequasi-periodicsubsystemof� � �for

J  �JDÍ 9oftenis referredto asa family of quasi-periodicattractors[62].

Also in thegeneralcaseresonancetonguesare(generically)presentandwe findthesameintricateinterplayof periodicityandquasi-periodicityasin theArnoldfamily.

Higherdimensionalanaloguesof thepresentsituationexist, wherenext to period-icity andquasi-periodicity, alsochaoticdynamicscoexists[19, 18]. Thisscenarioand the transitionsor bifurcationsbetweenthe variouskinds of dynamics,hasbeenassociatedto theonsetof turbulencein fluid dynamics,seeRuelleandTak-enset al. [63, 56]. Thequasi-periodicstatethenis seenasintermediatebetweenveryorderlyandchaotic.Also see[36, 44, 45].

4 The twist theorem

Returningto theconservativesettingweconsideranannulusÎ Á � 2 with ‘polar’coordinates

�ªÏc�xÐn�ÑN}P E £kÒ*�where

Òis compact.We endow Î with thearea

form·I�zÓ3ÏÕÔÕÓ Ð!%

Considerasmooth(say � � ) map� � � Î � Î of theform� � �ªÏ#�xÐn�(�j�ªÏ���@���^©�"Ðn�~�xÐn��� l �" ��Ö� (6)

thatpreservesthearea·h%

Notethatfor × �Øthemapleavesthefamily of circlesÐk�MÙ¨ÚuÛ

invariantandagainthe problemis the persistenceof this family forJ  !J

small. We saythat the map�8­

is a (pure) twist mapif the mapÐkq� ^���Ðn�

is adiffeomorphism.We assumeDiophantineconditionsas before: For given constants\ÜZ @

andY�Z let J_^I; ` b Jed Yb f �for all rationals!i b % Via themap

^>�nÐ�q� ^©�"Ð&�this setis pulledbackto a subsetÎ À�Á Î � which is theproductof acircleandaCantorsetof largemeasure.

Theorem 2 In the above circumstancesassumethat YzZ is sufficiently small.

ThenforJ  �J

sufficiently small, thereexistsa smooth( � � ) transformationof the

8

annulus ' � � Î � Î � conjugatingthe restriction�8­�J Ý É to a subsystemof

� � %Moreover, ' dependssmoothlyon

 3%Thistheoremis knownasthe‘twist’ theoremandit wasfirst provenby J.K.Moser[50]. Thepresentformulationis closeto theorem1 and,concerningthesmooth-nessof ' � � thesamecommentsapplyagainhere.

Discussion

Regardingthesimilarity betweenthetheorems1 and2 onecansaythatin theorem2 therole of theparameter hasbeentakenby theactionvariable

�%In thesame

spirit asbefore,we concludethat for the areapreservingcase,typically quasi-periodicityoccurswith positivemeasurein phasespace.

Remark A differencein thesettingsof theorem1 and2 is the following. In thecaseof KAM theorem1 for circlemaps,theconjugation' � betweenthefamily ofpurerotations

�®­andtheperturbation

� �preservestheprojectionto theparameter

space¥L&�u9¨§!�ØÞ�^#ße�

comparewith theequation(5). In thepresentcaseof theorem2 the correspondingprojection to the action spaceà � ÞgÐFß

generallyis notpreserved.

As anapplicationconsiderthemathematical(planar)pendulumwith (weak)peri-odic forcing. As possibleequationsof motiononemaytakeáâ �kã 2 Ê ��� â �  �ä¨å�ÊBÛ

oráâ �4�«ã 2 �= ¬äWå�ÊnÛG�&Ê ��� â � &�which give riseto volume-preserving3-dimensionalvectorfields. For simplicityweonly write down thefirst exampleæâ � �æ� � ;¤ã 2 Ê ��� â �= ¬äWå�ÊBÛæÛS� 9�%As is commonin mechanics,e.g., see[7], we introduceaction-anglevariables��Ð��ÖÏ��

for �jn�

i.e., for theautonomousplanarpendulum.In fact,denotingtheenergy by ç ­�� â ��� �(� E2 � 2 ;èã 2 ä¨å�Ê â � werestrictourselvesto theoscillatoryregionwhere ç ­g� â ��� �tH�ã 2 % Next considerany level set ç ­g� â �x� �é�Qê��

withJLê¬J¹H�ã 2 %

TheactionvariableÐ

thenis definedbyÐÂ�ªê��(� 9@g�]ëBì ²¢í�î � ï:ðòñBó ��Ó â �9

which is proportionalto theareaenclosedby the level set. TheanglevariableÏ

is obtainedby taking the time parametrizationof theperiodicmotion insidethislevel setscaledto period

@g�(%Thusoneobtainsthefollowing canonicalequationsæÐ��z&� æϸ�$^���Ðn�

for theoscillatorymotionsof theplanarpendulum.This impliesthatthePoincare (or stroboscopic)map

� �hastheaboveform (6). A

directcomputation3 shows that�8­

is a puretwist map. Thereforetheconclusionof quasi-periodicityoccurringwith positivemeasurein phasespace,applieshere.

A relatedapplicationdealswith coupledoscillatorsáâ E � ;¤ã 2E Ê ��� â E ;* ¬ô µô â E � â E � â 2 �áâ 2 � ;¤ã 22 Ê ��� â 2 ;*  ô µô â 2 � â E � â 2 �~�leadingto a4-dimensionalHamiltonianvectorfieldæâ / � � /æ� / � ;¤ã 2/ Ê ��� â E ;*  ô µô â / � â E � â 2 �Ö�õ �º9��Â@n�

with Hamilton function ç � � â E �x� E � â 2 ��� 2 �r� E2 � 2E � E2 � 22 ;aã E äWå�Ê â E ;ã 2 ä¨å�Ê â 2 �+  µr� â E � â 2 �~% In this caseit is the iso-energetic Poincare map whichobtainsthe form

� � %This leadsto the conclusionof quasi-periodicityoccurring

with positivemeasurein energy hypersurfacesof ç � %5 Conservative KAM theory

Theabove discussionhassetthestagefor a generalformulationof the KAM the-orem,asasuitablevariationon Kolmogorov [40, 41] andArnold [1].

We begin by introducingthenotionof a quasi-periodicinvarianttorusfor anau-tonomoussystemof ordinarydifferentialequations.Indeed,we understandthisto beaninvarianttorus,thedynamicson which is smoothlyconjugatedto thatofa constantvectorfield æ� E � ã Eæ� 2 � ã 2%u%u%Ü%u%W%o%u%u%æ� A � ã A �

3Involving anelliptic integral ö:öGö10

on thestandard� -torusP A � � A i ��@��¬÷ A �<�ËÞw� E �¢� 2 �W%u%u%W�G� A ße� wheretheangles� /

arecountedmodulo@��(�n9�ø õ ø � % Moreover, thefrequencies

ã E �Gã 2 �u%W%u%W�Gã Aare requiredto be independentover the rationals,which implies that the torusis denselyfilled by eachof its trajectories. In that casewe speakof � quasi-periodicity, bothfor therestrictionof thesystemto the � -torusasfor the � -torusitself.For integrableHamiltoniansystems,theconjugationwith aconstantvectorfield isprovidedby theactionanglevariablesprovidedby theLiouville-Arnold theorem[7].

Thepresentformulationof theclassicalKAM theoremnow reads

Theorem 3 In Hamiltoniansystemswith � degreesof freedomtypically � quasi-periodicLagrangeantori occurwith positiveLiouville measure,both

1. in phasespace;

2. andin eachenergy hypersurface.

As saidearlier‘typical’ meansthatthereareclassesof examplesthatare‘openinfunctionspace’.Theseexamplesarecloseto Liouville integrablesystems[7].Thefirst conclusionof theorem3 is thestandardKAM theorem,while thesecondis referredto astheiso-energeticKAM theorem.Thenondegeneracy conditionontheintegrableapproximationç � ç ��Ðn� is somewhatdifferentfor bothcases.Indeed,for thestandardKAM theoremtheKolmogorov nondegeneracy conditionis requiredwhichstatesthatthefrequency vectorã=� ô�çô Ðasa functionof theactionvariables

Ðshouldhavemaximalrank;this impliesthat

thefrequency mapÐ�q��ãé�"Ðn�

locally is a diffeomorphism.The condition for iso-energetic nondegeneracy similarly requiresthat the cor-respondingfrequency ratio map

ÐQq� �«ã E �"Ð&�0��ã 2 �"Ðn�z�Õ%u%u%Õ��ã A ��Ðn�¢� to the� � ;49g� -dimensionalprojective spaceshouldhave maximalrank. Comparewith[7, 19, 16]. We remarkthat theconclusiondrawn in theexampleof thetwo cou-pledoscillatorsof theprevioussectionalsowould follow asanapplicationof theiso-energeticKAM theorem.In typical cases,however, bothnondegeneracy conditionsaresatisfied,implyingthat theunionof quasi-periodicLagrangeaninvarianttori haspostivemeasureinphasespace,in sucha way that theconditionalmeasurein theenergy hypersur-facesalsois positive.

11

In additionto nondegeneracy oneagainneedsDiophantineconditionson thefre-quencies,in thiscasedefinedasfollows. For constants\]Z � ;a9 and Y�Z n� it isrequiredthatfor all ù N*÷ Ayú Þ�nß onehasJüû"ã�� ù&ý J d Y J ù JÃC f %Here

û"ã�� ùBý � . A/Gñ E ã / ù / � whileJ ù J®� . A/¢ñ E J ù / Jþ% Similar to the casesmet be-

fore, the correspondingsubsetof � A is nowheredenseof positive measure.Infact this Diophantinesubsetis a ‘bundle’ of closedhalf-lines,the intersectionofwhich with theunit sphereÿ A�CFE � for small Y � containsa Cantorsetof almostfullmeasure.Comparewith, e.g.,[23, 18].In thecaseof thegeneralKolmogorov nondegeneracy, thissetis pulledbackalongthe frequency mapin a locally diffeomorphicway. Iso-energeticnondegeneracymeansthat theenergy hypersurfacesaretransversalto theDiophantineline bun-dle,comparewith [19, 16].

Some applications of the KAM theorem

In theintroductionwe alreadymentionedpossibleapplicationsof theorem3. In-deed,in many modelsof classicalmechanicsonefindsLagrangeanKAM tori withquasi-periodicdynamics.As a ‘philosophical’applicationconsiderthesolarsystem,first takenasa pertur-bationof theintegrablesystemobtainedwhenneglectingtheinteractionbetweentheplanets.If the theoremwould apply it would follow that theactualevolutionhaspositive probability to bequasi-periodic,whenassumingthat the initial con-dition hasbeenchosenat random.In thatcaseonemight be inclined to call thesolarsystemstable.Much hasbeensaidaboutthis example[2, 53, 54, 64, 27] andherewe just re-strict to a few remarks.Firstly thesolarsystemcontainsquitestrongresonances,which necessitatea moresuitableintegrableapproximationthandescribedhere[2]. Secondlytheinteractionbetweentheplanetsprobablyis far toostrongfor anactualapplicationof theorem3 asaperturbationresult.Thethird remarkreferstorecentnumericalwork of Laskar[46], which seemsto show thatthesolarsystemis entirelychaotic,mankindjust doesnot exist longenoughto havenoticed

%u%u%¨%A large classof examplesis provided by consideringthe neighbourhoodof anygenericelliptic equilibriumpoint of aHamiltoniansystem.Undera few nonreso-nanceconditionsontheeigenvaluesof thelinearpartonecannormalizeanumberof stepsaccordingto Birkhoff, therebymakingthelowerorderterms� -torussym-metric. In factthisallowswriting thesystemlocally asanearlyintegrablesystem,wherethesizeof theperturbationis controlledby thedistanceto theequilibrium,compare[53, 54].

12

As a consequence,in a neighbourhoodof the equilibrium therearemany KAM

tori. Their union is of positive Liouville measure,wheretheequilibriumpoint isevena densitypoint of quasi-periodicity[58, 18].

Otherapplicationsof KAM-techiquesoccurin quantummechanics,in particularin thestudyof electron– Anderson– localization. If oneconsidersSchrodingerequationswith spatiallyergodic potentials,a localized(non-conducting)stateisan eigenfunctionof the Schrodingerequationat someenergy eigenvalue. Sucha function will decayexponentially in space. As in the localized regime theSchrodingeroperatortypically hasadensepointspectrum,developingaperturba-tion theoryof thecorrespondingresolventoperator, which divergesat this densesetof eigenvalues,runsinto problemsverysimilar to problemsencounteredin thetraditionalKAM set-up,wherethedensesetof resonancesgivesrise to divergentperurbationexpansions.KAM-inspiredproofsandanalysesof localizationhavebeendevelopedfor typical realizationsof randompotentialsin arbitrarydimen-sionsfor large interactionstrengthsor high energiesin arbitrarydimensions,seee.g. [30], aswell as for quasi-periodicpotentialsin onedimension,describingelectronsin quasicrystals,seee.g.[28, 20,61].Yet anotherfield of physicswhich is notoriousfor divergentperturbationtheoryproblemsandwhereKAM-like ideasarestartingto play a role is QuantumFieldTheory[33, 10].

Discussion

Therearea greatmany relatedissuesspreadover a vastliterature,comparewith[7, 8, 53, 54,64, 27], andwe will berestrictivehere.Onesubjectto mentionis the differencebetweenthe cases� �S@

and � d��&%Indeed,for � � @

the level setof the energy function is 3 dimensionalandtheLagrangeantori have dimension2. In the nearlyintegrablecasetheseKAM torigenerallyfoliate over nowheredensesets.However, for opensetsof initial con-ditions the evolution curvesareforever trappedin betweenthem,which impliesperpetualadiabaticstability.In contrast,for � d��

the Lagrangeantori have codimensionlarger than1 andevolution curvesmay escape.This actuallyoccursin caseof Arnold diffusion[4, 8, 24].Anothersubjectconcernsthe generalmotion in the neighbourhoodof the KAM

tori. Indeedby refinedaveragingit follows that nearbyevolution curves staynearbyover anexponentiallylong time, seeNekhoroshev [55]. Colloquially onesaysthattheKAM tori are‘sticky’.

Fromthefirst developmentsof KAM theoryon, it wasknown that thetheoryex-tendsoutsidethe Hamiltonianframe work of Lagrangeantori. So it turns out

13

that analoguesof theorem3 exist, e.g.,for lower dimensional(isotropic)invari-ant tori in Hamiltoniansystems.Here,apartfrom the internalfrequenciesof thequasi-periodictori, alsonormalfrequenciesplay a role andthesemoreover entertheDiophantineconditions.For a discussionon thedynamicalrole of thecorre-spondingnormal-internalresonancessee[13].Therehasbeenquitea lot of interestin theHamiltonianKAM theoryof normallyelliptic tori [59,38]. Thisalsoincludescasesin whichthenormalfrequencieshavestrongresonances[37]. Moreover, caseshave beenstudiedof normallyparabolictori. Thelattertwo casesinvolvequasi-periodictorusbifurcations,in certaincasesalsobranchingoff of higherdimensiontori [18, 34,14,15].

Similar resultsgo when consideringreversiblesystemsinsteadof Hamiltonianones[51,54,65, 17,22], or systemsthatareequivariantwith respectto certainLiegroups.Moreover, assection3 alreadyindicates,therealsoexist KAM theoremsfor thegeneralclassof (dissipative) dynamicalsystems,comparewith section3.In fact a unifying Lie algebraapproachallows to reachall theseresultsat once.In generaloneneedsexternalparametersto have a goodpersistencetheory forquasi-periodicinvarianttori. Comparewith [51, 52, 19,18].

6 Ergodic hypothesis: a paradox?

As adirectcorollaryof theorem3 wehave

Theorem 4 In typical Hamiltoniansystemswith � H��degreesof freedom,

energy hypersurfacesarenotergodicundertheHamiltonianflow.

Indeed,by theorem3 thereexist openclassesof Hamiltoniansystemswith in-varianttori, suchthat the union of thesetori haspositive (Liouville) measureinthe energy hypersurfaces.This leadsto a decompositionof thesehypersurfacesin severaldisjoint invariantsets,whichcontradictsergodicity. For backgroundonergodictheory, e.g.,see[8].

SotheKAM tori form anobstructionagainstergodicity andaquestionis how badthisobstructionis as � � �$% Thisquestionis of importancein statisticalphysics,wherelargesystemsarestudied.A primitive formulationof the ergodic hypothesisassertsthat typically, energyhypersurfacesof many-particlesystemsareergodic,which clearlywould not becompatiblewith theorem4. To fix thoughtswegiveanexample.

A lattice system

Considerthe1-dimensionallattice÷�� � � at theverticesof which identicalnon-

linearoscillatorsaresituated.For simplicity think of thelatticebeingsituatedon

14

a horizontalline, whereat all verticesidenticalpendulaaresuspended,subjecttoconstantverticalgravity. Also we connectnearestneighbouroscillatorsby weaksprings.Herethespringconstantscanbeeitherconstantor decayat infinity.Let � �aÁ�÷ betheboxwith vertices

� ; � � � �Ö% Then,for � ø � considertwo oftheseboxes �� Á � �<% We ‘truncate’ theinfinite systemby ignoringall pendulaoutsidethelargerbox � �¤%First considerthe integrablesystemassociatedto � ��� whereall interactionsareneglected.Supposethat the oscillatorssituatedat verticesin �� arein motion,while the othersareat rest. In the phasespacethis correspondsto a

@ � �j9-

dimensionalinvariant torus, which is normally elliptic. Moreover, the normalfrequenciesarein

9t�F9t�F%W%u%B�F9resonance.

Thenwe ‘turn on’ the activity of the interactingsprings. A suitableadaptation[37] of theorem3 andof [59] for this case,yieldsthepersistenceof theseelliptictori for small valuesof the springconstants.The correspondingkind of motionis a quasi-periodicbreather;for a similar kind of motioncomparewith [48]. Theunionof elliptic tori haspositive

@B�ª@ � �09g�-dimensionalHausdorff measure.

The questionnow is what is the asymptoticsof the ‘density’ of this measureas� (and � ��� �$%A partial answerto this question[37], saysthat this density

decaysexponentiallyfastin � � while thereis a furtherpolynomialdecayin � .

Remark It is worth mentioningthat thenormal9��(9���%u%W%��(9

resonanceof thissystemgivesriseto interestingquasi-periodicbranchings[37].

Discussion

The conclusionfrom this exampleis that the obstructionto ergodicity given byKAM theoryis not toobadasthesizeof thesystemtendsto infinity. This is in thesameline asanearlierresultby Arnold [3].

Anotherproblemis whathappensif thelimit � � �is reallyattained.TheKAM

theory for infinite systemsis fully in development[42, 39], but infinite latticesystemsstill havemany secrets.

7 Conclusion

The lastinginfluenceof Kolmogorov, Arnold andMoseron the presentstateoftheart in mathematics,physicsandothersciencesis enormousandthispaperonlysketchespartof this legacy. Neverthelesswe believe that it is an importantpart,whichis still fully in development.Theroleof theergodichypothesisin statisticalmechanicshasturnedout to bemuchmoresubtlethanwasexpected,seee.g.,[9,

15

31, 32]. RegardingKAM theory, for furtherreadingwe mentionthe introductorytexts [23, 49, 47,60]. Also readingof [18], [43] and[29] is recommended.TheauthorthankstheUniversitatdeBarcelonaandtheUniversitedeBourgognefor hospitalityduringthepreparationof thispaper. Also I thankAernoutvanEnterandFloris Takensfor helpfuldiscussion.

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