Ergodic Properties of a Class of Discrete Abelian

arXiv:0803.4044v1[math.DS]28Mar2008ErgodicPropertiesofaClassofDiscreteAbelianGroupExtensionsofRank-OneTransformationsChrisDodd∗PhakawaJeasakul†AnneJirapattanakul‡DanielM.Kane§BeckyRobinson¶NoahSteinkCesarE.Silva∗∗March28,2008AbstractWedefineaclassofdiscreteabeliangroupextensionsofrank-onetrans-formationsandestablishnecessaryandsufficientconditionsfortheseex-tensionstobepowerweaklymixing.Weshowthatallmembersofthisclassaremultiplyrecurrent.Wethenstudyconditionssufficientforshow-ingthatcartesianproductsoftransformationsareconservativeforaclassofinvertibleinfinitemeasure-preservingtransformationsandprovideex-amplesofthesetransformations.1IntroductionGroupextensionsofmeasure-preservingdynamicalsystemshavereceivedmuchattentionintheliterature.Inmostoftheworksthegrouphasbeenassumedtobecompact,andifthebasetransformationisfinitemeasure-preservingthentheextensionisfinitemeasure-preserving.Aquestionthathasbeenstudiedinthiscontextisconditionsunderwhichdynamicalpropertiesofthebasetransforma-tion(suchasweakmixingormixing)lifttothegroupextension;thereadermayrefertoe.g.[14],[13]andthereferencesintheseworks.Inthisarticlewe∗DepartmentofMathematics,MassachusettsInstituteofTechnology,MA,02139,[e-mail]†EconomicsDepartment,UniversityofCalifornia,Berkeley,CA94720,[e-mail]‡1022InternationalAffairsBuilding,ColumbiaUniversity,420West118thStreet,NewYork,NY10027,[e-mail]§DepartmentofMathematics,HarvardUniversity,Cambridge,MA02139,[e-mail]¶WilliamsCollege,Williamstown,MA01267,[e-mail]tionandDecisionSystems,MassachusettsInstituteofTechnology,Cambridge,MA02139,[e-mail]∗∗DepartmentofMathematics,WilliamsCollege,Williamstown,MA01267,[e-mail]1considerextensionsofaclassofrank-onetransformationsbycountablediscreteabeliangroups.Whilethebasetransformationisrestrictedtobearank-onetransformationweallowthegrouptopossiblybeinfinite.Weestablishasimpleconditionthatisequivalenttotheergodicityoftheextensions,andanothercon-ditionthatisequivalenttopowerweakmixingoftheextensions.Powerweakmixingisequivalenttoweakmixingforfinitemeasure-preservingtransforma-tions,butitisastrongerpropertyinthecaseofinfinitemeasure-preservingtransformations.Weshowthattheextensionispowerweaklymixingifitistotallyergodic.Wealsoshowthatourgroupextensionsaremultiplyrecurrent,andgiveseveralapplicationsshowingergodicityor(power)weakmixingforcertainextensionsinboththefiniteandinfinitemeasure-preservingcases.Inthelatersectionsweconsiderthequestionoftheconservativityofproductsofpowersofinfinitemeasure-preservingtransformations,andapplyourresultstostaircasetransformations.Let(X,B,μ)beameasurespaceisomorphictoafiniteorinfiniteinter-valinRwithLebesguemeasureμ(whentheintervalisfiniteweassumeμhasbeennormalizedtobeaprobabilitymeasure).LetT:X→Xbeaninvertiblemeasure-preservingtransformation.ThetransformationTisconser-vativeifforanysetAofpositivemeasure,thereexistsanintegeri0suchthatμ(T−iATA)0.TisergodicifforanypairofsetssetAandBofpositivemeasure,thereexistsanintegeri≥0suchthatμ(T−iATB)0.(Asourtransformationsareinvertibleanddefinedonnonatomicspaces,ergodicityimpliesconservativity.)LetT⊗ddenotethecartesianproductofd0copiesofT.WesaythatThasinfiniteconservativeindexifT⊗disconservativewithrespecttoμdforalld0(whereμddenotesd-dimensionalproductofμ);Thasinfiniteergodicindexifforalld0,T⊗disergodicwithrespecttoμd.AtransformationThaspowerconservativeindexifforallsequencesofpositiveintegersk1,k2,...kd,Tk1×Tk2×...×Tkd:X⊗d→X⊗discon-servative;Tissaidtobepowerweaklymixingifforallnonzerok1,...,kd,Tk1×Tk2×...×Tkdisergodic.Powerweakmixingisclearlyequivalenttoweakmixingforfinitemeasure-preservingtransformations,butitisastrongerpropertyinthecaseofinfinitemeasure-preservingtransformations[4].Infact,thereexistsatransformationT1suchthatT1hasinfiniteergodicindexbutT1×T21isnotconservative,hencenotergodic([4]).InSection2wedefine,foreachcountablediscreteabeliangroupG,aclassofmeasure-preservingtransformations.WhenGisaninfinitegroupthetransfor-mationisinfinitemeasure-preserving.AsthelastexampleinSection6shows,thesecontaingroupextensionsofrank-onetransformations.InTheorem2.2wegivenecessaryandsufficientconditionsforourconstructiontobepowerweaklymixing.WhenGisafinitegroup,thetransformationisfinitemeasure-preservingandourtheoremgivesequivalentconditionsforweakmixing.Wealsoshowthatourgroupextensionsaremultiplyrecurrent.Atransfor-mationTissaidtobed-recurrentifforallsetsofpositivemeasureAthereexistsanintegern0suchthatμ(A∩Tn(A)∩···∩Tnd(A))0.Tissaidtobemultiplyrecurrentifitisd-recurrentforallintegersd0.Asiswell-2known,Furstenbergshowedthateveryfinitemeasure-preservingtransformationismultiplyrecurrent[9],butitisnowknownthatinfinitemeasure-preservingtransformationsneednotbemultiplyrecurrent[8],[2],evenwhentheyarepowerweaklymixing[10].However,itwasshownrecentlythatcompactgroup[11]andσ-finite[12]extensionsofmultiplyrecurrentinfinitemeasure-preservingtransformationaremultiplyrecurrent,.Thisneednotbethecaseforexten-sionsbynon-compactgroup