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1、PeriodicPointsandTopologicalEntropyofOneDimensionalMapsLouisBlockJohnGuckenheimer*MichalMisiurewiczLaiSangYoung*Thispaperisadescendantofthestrikingtheoremof~arkovskiT[~a]aboutcontinuousmapsoftheinterval,statedhereasTheorem2.4.Webothgiveasimplerproofofthistheoremandextendittomapsof
2、thecircle.Letf:M÷Mbeacontinuousmapofaonedimensionalmanifoldintoitself.Weshallsayx(Mhasperiodnifxisaperiodicpointofminimalperiodn:fn(x)=xbutfi(x)~xforOin.Ourfirstproblemistodetermine,forwhichm,theexistenceofpointsofperiodsnIandn2forfimpliestheexistenceofpointsofperiodm.Weanswerthis
3、questioninalmostallcasesforwhichnI=I.ThesecondproblemistoobtainaminimalestimateforthetopologicalentropyoffusingonlytheinformationthatfhaspointsofperiodsnIandn2.AgainwegivedefinitiveanswersinthecasenI=I.Bothofthesequest4onshavebeenansweredpreviouslyformapsoftheinterval[~a,~t,M-S,J-
4、R]buttheresultsformapsofthecircleappeartobenew.Ourtechniquealsoappearstoapproachminimalsimplicityfortheproofsoftheseresults.Itreliesupontheconceptsoff-coversofsubintervalsandtheA-graphoffassociatedtoapartitionofM[B,B-F].Usingtheexistenceofpointswithselectedperiods,wefindapartition
5、AsuchthattheA-graphoffcontainsaparticularsubgraph.Theexistenceofotherperiodicorbitscanbededucedfromthesubgraph.Withausefullemmaforcalculatingthecharacteristicpolynomialofcertainmatrices,wealsoobtainestimatesfortopologicalentropy.Minimalmodelsformapswithfixedpointsshowthatourestima
6、tesaresharp.ThispaperowesitsexistencetotheInternationalConferenceonDynamicalSystemsheldatNorthwesternUniversityinJune1979.Wewouldliketothanktheorganizersofthesymposiumforarrangingsuchastimulatingmeetingaswellas*ResearchpartiallysupportedbytheNationalScienceFoundation.19theNational
7、ScienceFoundationandNorthwesternUniversityfortheirsupport.EvanstonmaynotbeRome,butithasbeenacenterforthedevelopmentofthisfieldofmathematics.WewouldliketodedicatethispapertoRufusBowenandPeterStefanwhoeachstudiedonedimensionalmapsbeforetheiruntimelydeaths.I.One-dimensionalmapsandgra
8、phsLetIbeaninterval,S1-acircle({z
