Dynamics, Laplace transform and Spectral geometry

arXiv:math/0508216v1[math.DG]12Aug2005DYNAMICS,LAPLACETRANSFORMANDSPECTRALGEOMETRY∗DANBURGHELEAANDSTEFANHALLERAbstract.WeconsideravectorfieldXonaclosedmanifoldwhichadmitsaLyapunovoneform.WeassumeXhasMorsetypezeros,satisfiestheMorse–Smaletransversalityconditionandhasnon-degenerateclosedtrajectoriesonly.Foraclosedoneformη,consideredasflatconnectiononthetriviallinebun-dle,thedifferentialoftheMorsecomplexformallyassociatedtoXandηisgivenbyinfiniteseries.Weintroducetheexponentialgrowthconditionandshowthatitguaranteesthattheseseriesconvergeabsolutelyforanon-trivialsetofη.MoreovertheexponentialgrowthconditionguaranteesthatwehaveanintegrationhomomorphismfromthedeRhamcomplextotheMorsecom-plex.Weshowthattheintegrationinducesanisomorphismincohomologyforgenericη.Moreover,wedefineacomplexvaluedRay–Singerkindoftorsionoftheintegrationhomomorphism,andcomputeitintermsofzetafunctionsofclosedtrajectoriesofX.Finally,weshowthatthesetofvectorfieldssatisfyingtheexponentialgrowthconditionisC0–dense.Contents1.Introduction22.Exponentialgrowth133.TheregularizationR(η,X,g)184.Completionoftrajectoryspacesandunstablemanifolds215.ProofofTheorem326AppendixA.ProofofProposition232AppendixB.VectorfieldsonM×[−1,1]33References34Date:February2,2008.2000MathematicsSubjectClassification.57R20,57R58,57R70,57Q10,58J52.Keywordsandphrases.Morse–Novikovtheory,Dirichletseries,Laplacetransform,closedtrajectories,exponentialgrowth,Lyapunovforms.∗Thispaperisanewversionofworkwhichhascirculatedaspreprint[6]underthename“Laplacetransform,dynamicsandspectralgeometry.”Themathematicalperspectiveofthisversionishoweververydifferent.PartofthisworkwasdonewhilethesecondauthorenjoyedthewarmhospitalityofTheOhioStateUniversity.ThesecondauthorwaspartiallysupportedbytheFondszurF¨orderungderwissenschaftlichenForschung(AustrianScienceFund),projectnumberP14195-MAT.PartofthisworkwascarriedoutwhenthefirstauthorauthorenjoyedthehospitalityofIHESinBuressurYvetteandsecondauthoroftheMaxPlanckInstituteforMathematicsinBonn.12DANBURGHELEAANDSTEFANHALLER1.IntroductionLetMbeaclosedsmoothmanifold.WeconsideravectorfieldXwhichadmitsaLyapunovform,seeDefinition3.WeassumeXhasMorsetypezeros,satisfiesMorse–Smaletransversalityandhasnon-degenerateclosedtrajectoriesonly.Theseassumptionsimplythatthenumberofinstantonsaswellasthenumberofclosedtrajectoriesinafixedhomotopyclassarefinite.Moreover,weassumethatXsat-isfiestheexponentialgrowthcondition,aconditiononthegrowthofthevolumeoftheunstablemanifoldsofX,seeDefinition4below.UsingatheoremofPajitnovweshowthatthesetofvectorfieldswiththesepropertiesisC0–dense,seeTheorem2.Letη∈Ω1(M;C)beaclosedoneform,andconsideritasaflatconnectiononthetrivialbundleM×C→M.UsingthezerosandinstantonsofXonemighttrytoassociateaMoretypecomplextoXandη.SincethenumberofinstantonsbetweenzerosofXisingeneralinfinite,thedifferentialinsuchacomplexisgivenbyinfiniteseries.Theexponentialgrowthconditionguaranteesthatthisseriesconvergesabsolutelyforanon-trivialsetofclosedoneformsη.FortheseηwethushaveaMorsecomplexC∗η(X;C),seesection1.5,which,asa‘function’ofη,canbeconsideredasthe‘Laplacetransform’oftheNovikovcomplex.TheexponentialgrowthconditionalsoguaranteesthatwehaveanintegrationhomomorphismIntη:Ω∗η(M;C)→C∗η(X;X),whereΩ∗η(M;C)denotesthedeRhamcomplexassociatedwiththeflatconnectionη.Itturnsoutthatthisintegrationhomomorphisminducesanisomorphismincohomology,forgenericη.TheseresultsarethecontentsofTheorem1andProposition12.ForthoseηforwhichIntηinducesanisomorphismincohomologywedefinethe(relative)torsionofIntηwiththehelpofzetaregularizeddeterminantsofLaplaciansinthespiritofRay–Singer.Ourtorsionhoweverisbasedonnon-positiveLapla-cians,iscomplexvalued,anddependsholomorphicallyonη.WhilethedefinitionrequiresthechoiceofaRiemannianmetriconMweaddanappropriatecorrectiontermwhichcausesourtorsiontobeindependentofthischoice,seeProposition14.CombiningresultsofHutchings–Lee,PajitnovandBismut–ZhangweshowthatthetorsionofIntηcoincideswiththe‘Laplacetransform’ofthecountingfunctionforclosedtrajectoriesofX,seeTheorem3.Implicitly,thesetofclosedoneformsηforwhichtheLaplacetransformofthecountingfunctionforclosedtrajectoriescon-vergesabsolutelyisnon-trivial,providingan(exponential)estimateonthegrowthofthenumberofclosedtrajectoriesineachhomologyclass,astheclassvariesinH1(M;Z)/Tor(H1(M;Z)).Moreover,thetorsionofIntηprovidesananalyticcon-tinuationofthisLaplacetransform,consideredasafunctiononthespaceofclosedoneforms,beyondthesetofηforwhichitisnaturallydefined.Therestofthepaperisorganizedasfollows.Theremainingpartofsection1con-tainsathoroughexplanationofthemainresultsincludingallnecessarydefinitions.Theproofsarepostponedtosections2through5andtwoappendices.1.1.Morse–Smalevectorfields.LetXbeasmoothvectorfieldonasmoothmanifoldMofdimensionn.Apointx∈MiscalledarestpointorazeroifX(x)=0.ThecollectionofthesepointswillbedenotedbyX:={x∈M|X(x)=0}.DYNAMICS,LAPLACETRANSFORMANDSPECTRALGEOMETRY3Recallthatarestpointx∈XissaidtobeofMorsetypeifthereexistcoordinates(x1,...,xn)centeredatxsothatX=Xi≤qxi∂∂xi−Xiqxi∂∂xi.(1)TheintegerqiscalledtheMorseindexofxanddenotedbyind(x).Arestpoi