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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
