measure theory of dynamical systems

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1、IntroductionTheinitialproblemsofergodictheoryaroseinadifferen-tiableframework:smoothflowsonenergymanifoldspreservingLiouvillemeasure.ItwasPoincar@whofirstintroducedpurelymeasuretheoreticconsiderations.Both,differentiableandmeasuretheoreticergodictheory,developpedtremendously,

2、especiallysinceKolmogorov'sdefinitionoftheentropyinvar-iant.Topologicalergodictheorygrewasanintermediarybetweenthetwofields.Itdealswithcontinuoustransforma-tionsandinvariantBorelmeasuresandcanbetracedbacktoamemoirofKrylov-Bogoliubov[127]in1937.Inthisvolumewetrytogiveasurveyof

3、thetopologicalergodictheoryoncompactspaces.Forconvenience,theunder-lyingspaceisassumedtobemetricandthetransformationtobeahomeomorphism.Eachoftheauthorshavinghisownpredilectioninthefieldandhisownstyle,theattempttowardsaunifyingapproachwassomehowhandicapped.Hopefully,the'essenc

4、e'will-however-bedistinguished.KrylovandBogoliubovprovedthattherealwaysexistinvariantmeasures.Ifthereisonlyoneinvariantmeasure,thesystemiscalleduniquelyergodic.TheprevalenceofthesesystemswasshownbyJewettandKrieger.Inthegeneralcaseitmightbepossibletosingleoutoneinvariantmeasur

5、ebysomevariationalprinciple.Ifthisisso,onehasanintrinsicallyergodicsystem.ImportantexamplesofthiskindarefoundbyParryandBowen.Both,uniquelyandin-trinsicallyergodic,systemswillbetreatedextensivelyinthesenotes.Inspiteofoureffortstocovertherecentpublicationsandtogivetherightperso

6、nscreditfortherighttheoremsweareawareofpossibleerrorsandwishtoapologizeforanysuchmistake.Alsowewishtoapologizeifsomebody'sworkcloselyconnectedwiththeaimofthissurveyisnotemphasizedasitshouldbe;inparticular,manyconcreteexamplesofdynamicalsystemsbearingalgebraic,combinatorial,di

7、fferentiableorotherstructuresarenotdiscussedhere.Wewouldliketosingleouttworeferences:TheLectureNotesofWalters[185]foranexcellentintroductiontoergodictheoryandofBowen[31]foradeepstudyofequi-libriumstatesandergodictheoryofimportantclassesofdifferentiabledynamicalsystems.Wehaveb

8、orrowedproofsfrommanysourcesandareparti-cularlyindebtedtoBowenandMis