The Ruelle Sullivan map for actions of Rn

Johannes Kellendonk1

Description

The Ruelle-Sullivan map for actions of Rn Johannes Kellendonk1 and Ian F. Putnam2,? 1 Institute Girard Desargues, Universite Claude Bernard Lyon 1, F-69622 Villeurbanne; e-mail: 2 Department of Mathematics and Statistics, University of Victoria, Victoria, B. C. V8W 3P4, Canada; e-mail: January 21, 2005 Abstract The Ruelle Sullivan map for an Rn-action on a compact metric space with invariant probability measure is a graded homomorphism between the integer Cech cohomology of the space and the exterior algebra of the dual of Rn. We investigate flows on tori to illuminate that it detects geometrical structure of the system. For actions arising from Delone sets of finite local complexity, the existence of canonical transversals and a formulation in terms of pattern equivariant func- tions lead to the result that the Ruelle Sullivan map is even a ring homomorphism provided the measure is ergodic. 1 Introduction We consider a variety of cohomology groups for a continuous Rn-action ? on a compact metric space X. Among them are the Cech cohomology Hˇ(X,Z) of X and the dynamical cohomology of the dynamical system (X,?) by which we mean the Lie-algebra cohomology H(Rn, C∞(X,R)) of Rn with coefficients in the ?-smooth real functions on X.

• invariant probability measure


• let µ


• probability measure


• ruelle-sullivan map


• cech cohomology


• defines canonical local


• continuously ?-differentiable


• rn-action ?

Sujets

• Desargues
• Probability measure
• ?ech cohomology

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 25
• Langue: English

Extrait

TheRuelle-Sullivanmapforactionsof R n JohannesKellendonk 1 andIanF.Putnam 2, ∗ 1 InstituteGirardDesargues,Universite´ClaudeBernardLyon1, F-69622Villeurbanne;e-mail:[e-mail] 2 DepartmentofMathematicsandStatistics,UniversityofVictoria, Victoria,B.C.V8W3P4,Canada;e-mail:[e-mail] January21,2005 Abstract TheRuelleSullivanmapforan R n -actiononacompactmetric spacewithinvariantprobabilitymeasureisagradedhomomorphism betweentheintegerCechcohomologyofthespaceandtheexterior algebraofthedualof R n .Weinvestigateflowsontoritoilluminate thatitdetectsgeometricalstructureofthesystem.Foractionsarising fromDelonesetsoffinitelocalcomplexity,theexistenceofcanonical transversalsandaformulationintermsofpatternequivariantfunc- tionsleadtotheresultthattheRuelleSullivanmapisevenaring homomorphismprovidedthemeasureisergodic. 1Introduction Weconsideravarietyofcohomologygroupsforacontinuous R n -actionϕon acompactmetricspaceX.AmongthemaretheCechcohomologyH ˇ (X, Z ) ofXandthedynamicalcohomologyofthedynamicalsystem(X,ϕ)by whichwemeantheLie-algebracohomologyH( R n ,C ∞ (X, R ))of R n with coefficientsintheϕ-smoothrealfunctionsonX.AnanalogoftheCech-de ∗ SupportedbytheNaturalSciencesandEngineeringResearchCouncilCanada 1

Rhamcomplexprovidesuswithagradedringhomomorphismbetweenthe .owt Cohomologycapturestopologicalinformationaboutthedynamicalsys- tem.IfwearealsogivenaninvariantBorelprobabilitymeasureµthen wecancapturesomeofthegeometricstructurewhichcomesfromthe(eu- clidean)geometryofthe R n -orbits.WedothisbyexploitingRuelle-Sullivan currentsassociatedwithsub-actions.Theseyieldlinearfunctionalsondy- namicalcohomologywhichwecombineintoagradedhomomorphismfrom thecohomologytotheexterioralgebraofthedual R n ∗ of R n .TheRuelle Sullivanmap τ ϕ,µ :H ˇ (X, Z ) → Λ R n ∗ (1) isdefinedasthecompositionofthetwohomomorphisms.Asimilargroup homomorphismrelatedtotheabovebutbetweentheK-groupK(C(X))and Λ R n ∗ canbefoundin[Co80]. OuraimhereistostudysomepropertiesoftheRuelle-Sullivanmapand toprovideinterestingandcomputableexampleswhichilluminatethatthe Ruelle-Sullivanmapdetectsgeometricalstructure.Forinstance,itcaptures thecontinuouseigenvaluesoftheaction(Theorem14)andcanbeusedto distinguishcut&projectionsetsoffixeddimensionandcodimension,which havethesamecohomologygroupbutprojectoutofgeometricallydifferent lattices(Theorem15). Alargepartofthepaperisdevotedtothemorespecialclassof R n - actionsarisingfromDelonesetsoffinitelocalcomplexityin R n .Inthis caseX=Ω P isthecontinuoushullofsuchaDelonesetP.Thissystem hastheparticularadvantagethatPalsodefinescanonicallocaltransversals forthe R n -action.ThesetransversalsareusedtodefineH( R n ,C tl ∞ c (Ω P , R )), thetransversallylocallyconstantdynamicalcohomologyofΩ P .Itarises fromasubcomplexoftheLie-algebracomplexandisisomorphictotheCech cohomologyH ˇ (Ω P , R )withrealcoefficientsundertheabovementionnedring homomorphism. Twofurthergroups,H ˇ P ( R n )andH ˇ ( R n ),thestronglyandweaklyP- equivariantcohomologygroupsof R n ,aredefinedascohomologiesofsub- complexesofthedeRhamcomplexfor R n .H ˇ P ( R n )isamoreintuitive pictureofH( R n ,C tl ∞ c (Ω P , R )).Theorems20,23containtheproofofthere- sultannouncedin[Ke03]thatthe(strongly)P-equivariantcohomologyis isomorphictotheCechcohomology(withrealcoefficients)ofthecontinu- oushullofP.H P ( R n )istheclosureofH ˇ P ( R n )inanappropriatesense 2