a bilateral version of the shannon-mcmillan-breiman theoremnew

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1、ABilateralVersionoftheShannon-McMillan-BreimanTheoremPierreTisseurLaboratoireG´enomeetInformatiqueUniversit´ed’Evry,TourEvry2.December5,2003AbstractWegiveanewversionoftheShannon-McMillan-Breimantheoreminthecaseofabijectiveaction.ForafinitepartitionαofacompactsetXandameasurableaction

2、TonX,wedenotebyCn,m,αT(x)theelementofthepartitionα∨T1α∨...∨Tmα∨T−1α∨...∨T−nαwhichcontainsapointx.Weprovethatforµ-almostallx,−1Tlimlogµ(Cn,m,α(x))=hµ(T,α),n+m→∞n+mwhereµisaT-ergodicprobabilitymeasureandhµ(T,α)isthemetricentropyofTwithrespecttothepartitionα.1IntroductionTheShannon-

3、McMillan-Breimantheorem[2],[3]isusedinmanyproblemsrelatedtothemetricentropymapofanergodicmeasure.Weextendthiswell-knownresulttothecaseofabijectivedynamicalsystem.OurprooffollowsthelineofPetersen’sproof[3].Weillustratethisnewresultwithanexamplethatgivesaninequalitybetweenshiftsandce

4、llularautomataentropiesandsomeanalogoftheLyapunovexponents.OurbilateralversionoftheShannon-McMillan-Breimantheoremisexpectedtobeusefulinotherareasofdynamicalsystems.12BackgroundmaterialLetXbeacompactspace,µaprobabilitymeasureonXandTameasurablemapfromXtoX.Wedenotebyαafinitepartitiono

5、fXandbyCT(x)n,αtheelementofthepartitionα∨T−1α∨...∨T−nαwhichcontainsthepointx.ForallpointxtheinformationmapIisdefinedbyXTI(α)(x)=−logµ(Cα(x))=−logµ(A)χA(x),A∈αwhereCT(x)istheelementofαwhichcontainsxandχisthecharacteristicαAfunctiondefinedby1ifx∈A,χA(x)=0otherwise.Theinformationmapsat

6、isfiesI(α∨β)=I(α)+I(β

7、α),(1)−1I(Tα

8、Tβ)=I(α

9、β)◦T.(2)ThesetwopropertiesareeasilyprovedfromthedefinitionofIandthefactthatTisasurjectivemap.Wereferto[3,p.238],[4,Chap.8]foradetailedproofof(1)and(2).AsimpleformulationofthemetricentropywithrespecttothepartitionαisgivenbyZnkhµ(T,α)=limI(α

10、∨

11、k=1Tα)(x)dµ(x),n→∞XwhereXµ(A∩B)I(α

12、β)(x)=−χA∩B(x)logµ(B)A∈α,B∈βistheconditionalinformationmaprepresentingthequantityofinformationgivenbythepartitionαknowingthepartitionβaboutthepointx.WerecalltheShannon-McMillan-Breimantheorem[2][3].Theorem1(Shannon-McMillan-Breiman’stheorem)Ifµi

13、saT-ergodicmeasure,thenfor