A simpler proof of the Poincar-Birkho theorem e

A simpler proof of the Poincar-Birkho theorem e

AsimplerproofofthePoincar´e-Birkho theorem

PatriceLeCalvezandJianWang

December17,2009

Introduction

Inhissearchforperiodicsolutionsintherestrictedthreebodyproblemofcelestialmechanics,H.Poincar´econstructedanarea-preservingsectionmapofanannulusAontheenergysurface.Heassertedthatanarea-preservinghomeomorphismoftheclosedannulusthatsatis essome“twistcondition”admitsatleasttwo xedpoints.Heproveditistrueinsomesimplecasesandconjectureditisalsotrueinageneralcase[5].Thenhedied.SowealsocallthetheoremthelastgeometrictheoremofPoincar´e.

In1913,Birkho [1]provedaresultwhichwasvalidto ndone xedpointbutuncorrecttogetthesecondone.Asmallmodi cationoftheargumentwasnecessaryandBirkho correctedthisminorerrorinapaper[2]publishedin1925.Seealsothewell-detailedexpositorypaperofBrownandNewman[3].

Thegoalofthisshortpaperistointroducethenotionofpositivepathofahomeo-morphismwhichseemstobeanaturalobjecttounderstandBirkho ’sarguments.Thisshortpaperisapartofthepaper[4].Onlyforcommunionwithinthemathe-maticsdepartmentofTsinghuaUniversity.

StatementandproofofthePoincar´e-Birkho theorem

InwhatfollowsapathonatopologicalspaceXisacontinuousmapγ:I→Xde nedonasegmentI=[a,b] R.Theoriginandtheextremityofγarerespectivelyγ(a)andγ(b).IfX1andX2aretwosubsetsofX,wewillsaythatγjoinsX1toX2ifitsoriginbelongstoX1anditsextremitybelongstoX2.TherestrictionofγtoacompactintervalJ Iisasub-pathofγ.Ifγisone-to-one,γisanarc;ifγ(a)=γ(b),itisaloop;ifγ(a)=γ(b)andγisone-to-oneon[a,b),itisasimpleloop.Theconcatenationof

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