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1、CHAPTER1VectorfieldsandC0topologyItiswellknownthatifavectorfieldvisclosetoavectorfieldwinC1topology,thentheintegralcurvesofvareclosetotheintegralcurvesofwinC1topology(seeforexample[28],Ch.10,§7).LesswellknownisthefactthatthepropertycitedaboveremainstrueifwereplaceC1topologybyC0topology.Astatementoft
2、hiskindcanbefoundforexamplein[13]§4“Continuity”.ThisC0-continuitypropertyisthemaintopicofthepresentchapter.Beforewestartletussettheterminology.ThetermmanifoldmeansC∞paracompactmanifoldwithoutboundary,havingacountablebase.Aclosedmanifoldisacompactmanifoldwithoutboundary.Thetermmanifoldwithboundary
3、orequivalently∂-manifoldmeansC∞paracompactmanifoldwithpossiblynon-emptyboundary,havingacountablebase(observethattheboundaryofa∂-manifoldmaybeempty).Theterm“smooth”isequivalentto“C∞”.1.Manifoldswithoutboundary1.1.Basicdefinitions.LetMbeaC∞manifoldandvaC1vectorfieldonM.AnyC1mapγ:I→Mdefinedonsomeopenin
4、tervalI⊂Randsatisfyingthedifferentialequationγ(t)=v(γ(t))foreveryt∈Iwillbecalledanintegralcurveofv.AnintegralcurveofviscalledmaximalifthereexistsnoextensionofγtoanintervalJwithIJ.Foreveryα∈Randeveryx∈Mthereisauniquemaximalintegralcurveofvsatisfyingtheinitialconditionγ(α)=x(thisfollowsfromthestan
5、dardtheoremsontheexistenceanduniquenessofsolutionsofdifferentialequations).IfMiscompacttheneverymaximalintegralcurveisdefinedonthewholeofR.18Chapter1.VectorfieldsThevalueattofthemaximalintegralcurveγsatisfyingγ(0)=xwillbedenotedbyγ(x,t;v).Forthecurveitselfweshallusethenotationγ(x,·;v).ForasubsetA⊂Rw
6、edenotebyγ(x,A;v)thesetofallpointsγ(x,t;v)witht∈A.Therestrictionofthemaximalintegralcurveγ(x,t;v)tothesubsetofallnon-negativet∈Iwillbecalledthetrajectoryofvorthev-trajectorystartingatx.AsubsetXofMiscalledv-invariant,ifa∈X⇒γ(a,t;v)∈Xforeveryt0.AsubsetXofMiscalled±v-invariant,ifitisv-invariantand(
7、−v)-invariant.Inotherwords,Xis±v-invariant,ifeverymaximalintegralcurvepassingthroughapointofXneverleavesX.Twomaximalintegralcurvesγ1,γ2ofvarecalledequivalentiftheycanbeobtainedonefromanotherbyreparameterization,thatis,
