Alcalde Cuesta P Gonzalez Sequeiros and A Lozano Rojo Departamento de Xeometra e Topoloxa Universidade de Santiago de Compostela Departamento de Didactica das Ciencias Experimentais Universidade de Santiago de Compostela Departamento de Matematica ID: 54518
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ii)Secondlyweprovethat@R1isR-thin[4],i.e.(@R1)=0foreveryR-invariantprobabilitymeasure.In[7],C.Serieshasprovedthatanyfoliationwithpolynomialgrowthishypernite.Here,wewillusethesameoutline(whichremindstheproofoftheRohlinlemma).iii)Finally,intheEuclideancase,wepovethatR1isminimalandeveryR-classsplitintoanitenumberofR1-classes.Thiswillallowustoconcludebyapplyingtheorem4.18of[4].Infact,thisproofappliestothebroaderclassoftilablelaminations[1]andwededucethefollowingresult(whichextendsthemaintheoremof[5]):Corollary.-AnyfreeminimalactionofZmontheCantorsetisa able.References[1]J.Bellissard,R.Benedetti,J.M.Gambaudo,SpacesofTilings,FiniteTelescopicApproximationsandGap-Labelling,Comm.Math.Phys.261(2006),1-41.[2]E.Ghys,LaminationsparsurfacesdeRiemann,Panor.Syntheses8(1999),49-95.[3]T.Giordano,I.Putnam,C.Skau,TopologicalorbitequivalenceandC-crossedproducts,J.reineangew.Math.469(1995),51-111.[4]T.Giordano,I.Putnam,C.Skau,A ableequivalencerelationsandorbitstructureofCantorminimalsystems,ErgodicTheoryDynam.Systems24(2004),441-475.[5]T.Giordano,I.Putnam,H.Matui,C.Skau,OrbitequivalenceforCantorminimalZ2-actions,preprint.[6]H.Matui,A abilityofequivalencerelationsarisingfromtwo-dimensionalsubsti-tutiontilings,ErgodicTheoryDynam.Systems26(2006),467-480.[7]C.Series,Foliationsofpolynomialgrowtharehypernite,IsraelJ.Math.34(1979),245-258.2