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1、CHAPTER13CountingclosedorbitsofthegradientflowInSections1and2weintroduceandstudytheLefschetzzetafunctionofthegradientflowofacircle-valuedMorsefunction.Thisfunctionisapowerserieswhichcontainsinformationabouttheclosedorbitsoftheflow.Themainresultofthischapter
2、(Theorem4.3)statesthattheLefschetzzetafunctioncanbeexpressedintermsofWhiteheadtorsioninvariants,associatedwiththeNovikovcomplex.WediscusstheWhiteheadtorsionandrelatedtopicsinSection3.1.LefschetzzetafunctionsofthegradientflowsDefinition1.1.LetwbeaC1vectorfie
3、ldonamanifoldM.Aclosedtrajectoryofwisanon-constantintegralcurveγ:[a,b]→Mofw,suchthatγ(a)=γ(b).Wesaythattwoclosedtrajectoriesγ:[a,b]→M,γ:[a,b]→Mareequivalent,ifthereisCsuchthata=a+C,b=b+Candγ(t+C)=γ(t)foreveryt.Aclosedorbitofwisaclassofequivalenceofcl
4、osedtrajectories.ThesetofallclosedorbitsofwisdenotedbyCl(w).Weshallidentifyaclosedtrajectorywiththecorrespondingclosedorbit,whennoconfusionispossible.Foraclosedorbitγ:[a,b]→Mletc∈[a,b]bethesmallestnumber>asuchthatγ(a)=γ(c).Itiseasytoseethatb−a=n(c−a)wher
5、enisanaturalnumber,calledthemultiplicityofγ,anddenotedmult(γ).Ifmult(γ)=1,theorbitγiscalledaprimeclosedorbit.ThesetofallprimeclosedorbitsofwisdenotedbyClPr(w).Letγ:[a,b]→Mbeaclosedtrajectory.LetΣbeanysubmanifoldofMwithdimΣ=dimM−1,containingp=γ(a)andsucht
6、hatwistransversetoΣatp(thatis,w(p)∈/TpΣ).TheclassicalconstructionofthePoincar´ereturnmapassociateswiththesedataadiffeomorphismR:U→V,whereU,VareneighbourhoodsofpinΣ.Hereisabriefdescriptionoftheconstruction(werefertothebookofJ.PalisandW.deMelo[122]formorede
7、tails):foreachpointx∈Σinasmallneighbourhoodofpthew-trajectorystartingatxwillintersectΣagainatsomepointyatsome384Chapter13.Countingclosedorbitsofthegradientflowmomentt0closetob;wesetbydefinitionR(x)=y.IfpisanisolatedfixedpointofR,thenitsindexiscalledtheindex
8、ofγ,anddenotedbyi(γ).Thenumberi(γ)iF(γ)=m(γ)iscalledtheFullerindexofγ.WesaythatγisahyperbolicclosedorbitifpisahyperbolicfixedpointofthePoincar´ereturnmap.Nowwewillconcentrateonthecaseofgradientsofcircle-valuedMorsefunctions
