RENORMALIZATION GROUP AND FIELD THEORETIC TECHNIQU

RENORMALIZATIONGROUPANDFIELDTHEORETICTECHNIQUESFORTHEANALYSISOFTHELINDSTEDTSERIESALBERTOBERRETTIANDGUIDOGENTILEAbstract.TheLindstedtserieswereintroducedintheXIXcenturyinAstronomytostudyperturbativelyquasi-periodicmotionsinCelestialMechanics.InMathematicalPhysics,aftergettingtheattentionofPoincar´e,whostudiedthemwidelybypursuingtoallorderstheanalysisofLindstedtandNewcomb,theirusewassomehowsupersededbyothermethodsusuallyreferredtoasKAMtheory.Onlyrecently,afterEliasson’swork,theyhavebeenreconsideredasatooltoproveKAM-typeresults,inaspiritclosetothatoftheRenormalizationGroupinquantumfieldtheory.FollowingthisnewapproachwediscussheretheuseoftheLindstedtseriesinthecontextofsomemodelproblems,likethestandardmapandnaturalgeneralizations,withparticularattentiontothepropertiesofanalyticityintheperturbativeparameter.1.IntroductionManyplanetarymotionsareapproximatelyperiodic.Outstandingexamplesaregivenbythespin-orbitresonancesbetweentherevolutionandtherotationsatelliteperiods(see[43])andbyLagrange’sequilateralsolutionsforthethreebodyproblem,asinthecaseofSun,JupiterandtheTrojangroup(see[56]).FromamathematicalpointofviewtheexistenceofperiodicsolutionsinCelestialMechanicsproblemswasprovedrathersoon;wecanrefertotheclassicalworksbyPoincar´e[53]andBirkhoff[16].Renewedinterestaroseaboutquasi-periodicmotionssincetheoriginaryformulationoftheKAMtheorem;seeforinstance[51]forareview.Theperturbativeseriesofthequasi-periodicsolutionswerewellknowninAstronomy:theywereintroducedandstudiedtofirstorders,independently,byLindstedt[47]andNewcomb[52],andhavebecomeknownasLindstedtseries.ThenPoincar´eshowedthattheserieswerewelldefinedtoallorders(seeforinstance[53]),butonlywiththeworkofKolmogorov[44]theexistenceofquasi-periodicmotions,hencetheconvergenceoftheseries,wasproved.SoonafterwardsnewproofsofKolmogorov’stheoremweregivenbyArnol’d[1]andbyMoser[50],whotreatedthenonanalyticcase.Successivelyalotofworksweredevotedtosuchafield,andgaverisetowhathassincethenreferredtoasKAMtheory:allsuchworksobtainedtheconvergenceoftheLindstedtseriesnotbydirectlyanalyzingtheseriesitself,butasabyproductoftheproofofexistenceofquasi-periodicsolutions.Recently1anewproofofKolmogorov’stheoremhasbeengivenbyEliassonbystudyingdirectlytheperturbativeseries,hencewithoutusingtheiterativerapidlyconvergentproceduretypicalofthestandardversionsofKAMtechniques.NotethattheapproachfollowedEliassonisquitenatural,mostlyfromaphysicalpointofview(andinfactitwasthefirsttobeintroduced):iflookingforquasi-periodicsolutionsthefirstattemptonecanthinkaboutistowriteformallythesolutionasaquasi-periodicfunction,withcoefficientstobedetermined,andinsertitintotheequationsofmotion,tocheckifthereissomechoiceofthecoefficientsforwhichtheequationsofmotionsaresatisfied.1Eliasson’sresultswereannouncedin1986[26],andpresentedtofullextentin1988inaReportoftheMathematicsDepartmentoftheUniversityofStockholm;thelatterwaspublishedonlyin1996[27].2ALBERTOBERRETTIANDGUIDOGENTILEHerewefollowEliasson’sapproachintheRenormalizationGroupinterpretationgivenby[29]anddevelopedinaseriesofsubsequentpapers;see[38,39]andthereview[40].Wedonotinsistonthewidefieldofapplicabilityofthemethod,byreferringtotheliteratureforadetaileddescriptionofalltheresultsthatweshallnotevenmentioninthispaper(see[32,30,36,33,34,35,17,18,19,3];areviewcanbefoundin[37]).Ratherweprefertodiscusssomespecialdynamicalsystems,thestandardmapandotherrelatedmodels,whichcapturemostofthegeneralinterestingfeatures,andtodescribeinsomedetailsaseriesoforiginalresultswhichwehaverecentlyobtainedwithsuchtechniques(see[7,8,9,10,11]).ThestandardmapisgivenbyTε:(x′=x+y+εsinx,y′=y+εsinx.andwasintroducedbyGreene[41]andChirikov[24]asaparadigmaticalexampleofdynamicalsystem:itissimpleenoughtokeepseparatethenon-trivialdynamicalfeaturesfromthetechnicalintricaciesofothermodelsofhigherdimension.Thestandardmaphasalsomoredirectreasonstobeinteresting.Infact,itcanbegeneratedformallybythehamiltonianH=12y2+εcosxXk∈Ze2πikt,whichisofcourseverysingular,sincethesumontherightgivesasumofδfunctionsatintegervaluesoft(hencethenameof“kickedrotator”whichisgivensometimestothemodel).Bytruncatingthesum(thatis,bykeepingonlythetermswithaslowlyvaryingangularvariablex),weobtainhamiltoniansoftheforcedpendulumtype,whichareremarkablysimilartothoseusedassimplifiedmodelsofthespin-orbitinteractioninCelestialMechanics.See[46]forthederivationofthehamiltonianHand[43]forthespin-orbitinteraction(seealso[21]fortheso-calledspin-orbitmodel).Thepaperisorganizedasfollows.InSection2weshalldefinethestandardmapandnaturalextensionsofit,whichweshallcallgeneralizedstandardmaps.Weshallintroducealsosomesimplifiedmodels,thesemistandardmapand,inthesamespirit,thegeneralizedsemistandardmaps.InSection3weshallstateourmainmathematicalresultsabouttheanalyticitypropertiesofbothperiodicandquasi-periodicsolutions.Inparticularweshallseethatitispossibletowritethesolutionsasfunctionsofsuitablevariablesintermsofwhichthedynamicsisatrivialrotation;weshallcallconjugatingfunctionsthefunctionswhichcarryoutsuchatask.Next,inSection4weshalldevelopthemathematicaltoolswhichsh