Aaronson, J.

Denker, M.

English

Pure OAI Repository

On exact group extensionsCitation for published version (APA):Aaronson, J., & Denker, M. (1999). On exact group extensions. (Report Eurandom; Vol. 99047). Eurandom.Document status and date:Published: 01/01/1999Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Report 99-047On Exact Group ExtensionsJon AaronsonManfred DenkerISSN 1389-2355ON EXACT GROUP EXTENSIONSJON AARONSON AND MANFRED DENKERABSTRACT. We give conditions for the exactness of JR.d-extensions.§O INTRODUCTIONA nonsingular transformation (X, B, m, T) of a standard probability space iscalled a fibred system if there is a generating measurable partition 0 such thatT : a ~ Ta is invertible, nonsingular for a E 0, and a Markov map (or Markovfibred system) if in addition, Ta E 0'(0) mod m \:j a E o.Write 0 = {as : s E S} and endow SN with its canonical (Polish) producttopology. Letn~ = {s = (81,82,.") E sN: m(n T-ka Sk ) > 0 \:j n:;::: I},k=1then ~ is a closed, shift invariant subset of SN, and there is a measurable map4> : ~ ~ X defined by {4>(st, S2, ..• )} := n~=1T-(k-l)a Sk 'The closed support of the probability m' = m 0 4>-1 is ~, and 4> is a conjugacyof (X,B,m,T) with (~,B(~),m',shift). Thus we may, and sometimes do, assumethat X = ~, T is the shift, and 0 = {[s] : s E S}.For n :;::: 1, there are m-nonsingular inverse branches of T denotedVa : Tn a ~ a and with Radon Nikodym derivatives denoted, dm 0 VaVa := dm .Let (X, B, m, R) be a nonsingular transformation of a standard probability space.The Frobenius-Perron operators PRn = PRn,m : L 1(m) ~ L 1(m) are defined byand for the locally invertible (X, B, m, T, 0) (as above) have the formPTnf = L ITn a V:' f 0 Va'aEQ~-l1991 Mathematics Subject Classification. Primary: 28D05, 60B15; Secondary: 58F15, 58F19,58F30.©July 1999, revision 4/10/99Typeset by A,;\IfS-'lEX12 JON AARONSON AND MANFRED DENKERA locally invertible map (X, B, m, T, a) has:the Renyi property if:J C> 1 such that V n ~ 1, a E a~-l, m(a) > 0:I~!~:~ I ::; C for m x m-a.e. (x, y) E Tna x Tn a.It is well known (a proof is recalled in [A-D-U]) that any topologically mixingprobability preserving Markov map with the Renyi property is exact in the sensethat nn>1 T-n B = {0, X} mod m.Examples include:• topological Markov shifts equipped with Gibbs measures ([Bo],[Bo-Ru]) and• uniformly expanding, piecewise onto C 2 interval maps T : [0, 1] ~ [0,1] satisfyingAdler's condition sUPxE[O,1] Ir~~)~1 < 00 ([Ad]);or, more generally,• Gibbs-Markov maps as in [A-D1].Now let ¢ : X ~ ]Rd be measurable and consider the skew product Tlj> : X x]Rd ~X x]Rd defined by Tlj>(x, y) := (Tx, y+¢(x)) with respect to the (invariant) productmeasure m x mRd where mRd denotes Lebesgue measure.We say that ¢ is aperiodic if ,(¢) = zhh 0 T has no nontrivial solution in , EJRd , Z E S1 and h : X ~ S1 measurable. It is not hard to show that if Tlj> is ergodic,and T is weakly mixing, then Tlj> is weakly mixing iff ¢ is aperiodic.We're interested in the exactness of Tlj>.We establish two (partial) results in this direction.Theorem 1.Suppose that (X, B, m, T, a) is a probability preserving Markov map with theRenyi property. Let N ~ 1 and ¢ : X ~ ]Rd be a~-1-measurable (i.e. ¢(x) =¢(a~-1(x)) where x E a~-1(x) E a~-1).If Tlj> is topologically mixing, then Tlj> is exact.For the other result, we assume that (X, B, m, T, a) is an exact probabilitypreserving locally invertible map with the property that for some Banach space(L, II· IlL) of functions with II ·112 ::; II . IlL, such that PT : L ~ Land :JM > 0, () E(0,1) such thatIIPTnf - i fdmllL ::; M(}nllfilL V f E L.This property can be obtained as a consequence of the quasi compactness of Doeblin-Fortet operators, see [D-F]' [IT-M]).Given ¢ : X ~ ]Rd measurable, we define the characteristic function operatorsPt(J) = PT(ei(t,lj» J) (t E JRd).We assume also that Pt : L ~ L (t E ]Rd) and that t 1-+ Pt is continuous(JRd ~ Hom(L, L).It is shown in [Nag] (see also theorem 4.1 of [A-Dl]) that(i) there are constants € > 0, K > °and () E (0,1); and continuous functions,X : B(O, €) ~ Bc(O, 1), g : B(O, €) ~ L such thatIlptnh - 'x(t)ng(t) Ix hdmllL ::; K(}nllhil L Viti < €, n ~ 1, hE L;andON EXACT GROUP EXTENSIONS(ii) in case ¢> is aperiodic, then '\f 0 < 0 < M < 00, :J K > 0, 0 : X _ ]Rd uniformlyHolder continuous on partition sets. Here L is a space of Holder continuous func-tions f : X-C.• (see [Rou], [Ry]), X = [0,1], m Lebesgue measure, Q a partition of X mod minto open intervals, and T : a - Ta an invertible, m-nonsingular homeomorphismfor each a E Q with inf IT'I > 1 and i, of bounded variation on X; and ¢> : X _ ]Rdeither: of bounded variation on X; or constant on each a E Q.Set ¢>n = ¢> + ¢> 0 T + ... + ¢> 0 Tn-I.Theorem 2.Suppose that(0) '\f ,X > 1 :J nk - 00 such that ¢>nk _ 0 a.e. as k - 00,Xnkand that ¢> is aperiodic;then Tel> is exact.Remarks.1) Theorem 2 generalises the corresponding theorem on page 443 in [G].2) The condition (0) is satisfied if m-dist (¢» is in the domain of attraction of astable law.3) The condition (0) is not satisfied iff :J ,X > 1 and c > 0 such that m([!¢>n I >,Xn]) ~ c '\f n ~ 1 and there are independent processes like this.§1 FROBENIUS-PERRON OPERATORS, EXACTNESS AND RELATIVE EXACTNESSLet (X, B, m, R) be a nonsingular transformation of a standard probability space.The tail (J-algebra of (X, B, m, R) is T(R) := n:=l R-nB and the nonsingulartransformation R is called exact if = {0, X} mod m.Theorem 1.1 [D-L].IIPRnflh -IIE(fIT(R))III as n - 00 '\f f E L1(m).In particular (see [L]), R is exact iff IIPRnflh - O'\f f E L1(m), Ix fdm = O.Proof.First note that IPT fl :::; PT!fl whence IIPRnfll1 1and:J limn _ oo IIPRnflll' Next,'\f n ~ 1 :J gn E LOO(B) with Ix(PRnf)gndm = IIPRnflll' whenceIIPRnfll1 = Lfgn 0 Rndm.By weak * compactness, :J nk - 00 and 9 E LOO(B) such that gnk 0 Rnk ---" 9 weak* in £OO(B).4 JON AARONSON AND MANFRED DENKERIt follows that g E Loo(T(R)), IIglloo ~ 1 and limn_>oo IIPRnfl11 = Ix fgdm.Thuslim IIPRnflh ~ sup { r fhdm: hE Loo(T(R)), Ilhll oo ~ I} = IIE(fIT(R))111'n->oo JXTo show the converse inequality, note that :3 g E Loo(T(R)), IIglloo = 1 suchthatIIE(fIT(R))lh = i E(fIT(R))gdm = i fgdmwhence V n ~ 1, :3 gn E Loo(B), g = gn 0 Rn andIIE(fIT(R))lh = i fgdm = i fgn 0 Rndm = i (PRnf)gndm ~ IIPRnflh·oLet (X, B, m, R) and (Y, C, jJ, S) be nonsingular transformations of standardprobability spaces. A factor map is a function 7r : X ----+ Y satisfying 7r-1C CB, 7r 0 T = So 7r, m 0 7r- 1 = jJ.The fibre expectation of the factor map 7r : X ----+ Y is an operatorf ~ E(fI7r), L1(X,B,m) ----+ L1(Y,C,jJ)defined by Iy E(fI7r)gdjJ = Ix fg 0 7rdm.The factor map 7r : X ----+ Y is called relatively exact iff E L1(B), E(fI7r) = 0 a.e. ===? IIPRnflh ----+ O.The corollary below appears in [G]. For the convenience of the reader, we supplya (possibly different) proof.Proposition 1.2. Suppose that 7r : X ----+ Y is relatively exact, then T(R) =7r- 1T(S) mod m.Proof.Evidently, 7r- 1T(S) ~ T(R). We show that 7r-1T(S) 2 T(R).By relative exactness and theorem 1.1, if f E L1(B) and E(f/7r) = 0 a.e., thenIx fgdm = 0 V g E Loo(T(R)).Thus if f E L2 (B) e L2 (7r- 1C), then E(fI7r) = 0 a.e. and soi fgdm = 0 V g E Loo(T(R)), ===? f..l L2 (T(R)).Thus L2 (B) e L2 (7r- 1C) c L2(B) e L2(T(R)) whence L2 (T(R)) C L2 (7r- 1C) andT(R) C 7r-1C mod m.To see that in fact T(R) ~ 7r-1T(S) mod m, fix N ~ 1, thenT(R) = nR-nB = n R-nBn~1 n~N+1= R-NT(R) C R-N7r-1C = 7r- 1S-NC.Taking the intersection over N shows the claim. 0Corollary 1.3 ([G], proposition 1).If S is exact and 7r : X ----+ Y is relatively exact, then T is exact.ON EXACT GROUP EXTENSIONS§2 PROOF OF THEOREM 1For a nonsingular transformation (X, B, m, R), define the tail relation of R:5Evidently 'r(R) is an equivalence relation and if (X, B, m) is standard, then'r(R) E B(X x X).If R is locally invertible, then 'r(R) has countable equivalence classes and isnonsingular in the sense that m('r(R)(A)) = °\f A E B, m(A) = °where'r(R)(A) := {y EX: :3 x E A (x, y) E 'r(R)}.A set A E B(X) is invariant under the equivalence relation 'r E B(X x X) if'r(A) = A and the equivalence relation 'r is called ergodic if'r-invariant sets haveeither zero, or full measure.The collection of invariant sets under 'r(R) is the tail a-algebra T(R) (whencethe name" tail relation").In order to prove theorem 1, it suffices to show that 'r(T<t» is ergodic.The tail relation of T<t> is given by'r(T<t> )= {((x, s), (y, t)) E (X X G)2: :3 n 2: 0, Tnx = Tny, S - t = cPn(Y) - cPn(x)}= {((x, S), (y, t)) E (X X G)2: (x, y) E'r(T), ¢(x, y) = S - t}where ¢ : 'r(T) -+ lRd is defined by ¢(x, y) := 'L':=o(cP(Tn y) - cP(Tn x )).We prove that 'r(T<t» is ergodic by the method of Schmidt (explained in [S]), byshowing that \f t E lRd , U a neighbourhood of t and A E B m(A) > 0, :3 B EBB cA and T : B -+ B nonsingular such that (x, T(X)) E 'r(T) and ¢(x, T(X)) E U \f x EB.This boils down to showing that\f A E B+ go E lRd TJ > 0, :3 B E B+ B c A, n 2: 1and T : B -+ T B c A nonsingular such that(+) Tn 0 T Tn and II cPn 0 T - cPn - go II < TJ on B.The proof of (+) will be written as a sequence of minor claims, '0,'1, ....'0 We first claim that there is no loss in generality in assuming that N = 1 (i.e.that cP : X -+ lRd is a-measurable). This is because (X, B, m, T, a~-l) is also aprobability preserving Markov map with the Renyi property and inducing the same(shift) topology on X as (X, B, m, T, a).'1 \f s, t E S, :3 '" = "'s,t 2: 1 and a = as,t = [a1, ... aK]' b = bs,t = [bI, ... bK] Ea~-l, a1 = b1 = S aK= bK= t such that IlcPK(b) - cPK(a) - goll < TJ.This follows from topological mixing of T<t>.By the Renyi property, :3 M > 1 such that6 JON AARONSON AND MANFRED DENKERGiven U = [Ub ... ' un] E Q~-1 with Un = t, define T = Tu : unT-na -+ unT-nbby'2 T = Tu : un T-n a -+ un T-nb is invertible nonsingular and d;;';T = M±4:~~~.PROOF1 dmOTd dm = m(unT-nbnc)unT-nanc m= M±2 m(b) m(u)m(b)m(c)m(a)= M±4 m(b) m(u n T-na n c).m(a)o'3 PROOF OF :j:Fix 0 < E < M-1 min{m(as ,d, m(bs,t)}, thenLet 8 > 0 be so small that 8 < m~~~(a~).:3 n ~ 1 and u E Q~-1 such that m(A n u) ~ (1- 8)m(u) and [8] e T[un].Consider Tu : un T-n a -+ un T-nb as in '2. Evidently Tn+K 0 T Tn+K andII<Pn+K 0 T - <Pn+K - goll < 'TJ on un T-na.To complete the proof we claim that :3 B E B+ B e An u n T-n a such thatTBeA.To see this, note thatm(u n T-na n A) ~ m(u n T-na) - m(u \ A) ~ (E - 8)m(u),whence using '2,m(b) m(b)(E - 8)m(T(u n T-na n A)) ~ M4m(a) m(u n T-na n A) ~ M4m (a) m(u).Since T(U n T-na n A) e u, the condition on 8 > 0 ensures that m(T(u n T-n a nA) n A) > 0 whence m(B) > 0 where B := T- 1 (T(U n T-n a n A) n A) eA. 0§3 PROOF OF THEOREM 2We prove theorem 2 via corollary 1.3. To do this, we must consider Ttl> as anonsingular transformation with respect to some probability P rv m x mJRd.Let p : JR.d -+ ~ be continuous with JJRd p(y)dy = 1 and define a probabilityP on X x JR.d by dP(x, y) := p(y)dm(x)dy; then (X x JR.d, B(X X JR.d), P, Ttl» is anonsingular transformation with Frobenius-Perron operators given byON EXACT GROUP EXTENSIONS1PTn p f (x, y) = -() PTn (J . 1@p)(x,y)q,' py q,7where PTn := PTn mXm d·q, q,' IRConsider the map 7r : X x ]Rd ---+ X defined by 7r(x, y) = x. This is a factor mapas it satisfies 7r- 1B(X) C B(X x ]Rd), 7r 0 T</> = To 7r, Po 7r-1 = m.The fibre expectation of 7r is given byE(JI7r)(x) = [ f(x, y)p(y)dy (J E L1(X X ]Rd, B(X X ]Rd), P)).J[?dBy corollary 1.3 and exactness of T, it suffices to show that 7r is relatively exact.To do this, we show that[ f(x, y)p(y)dy = 0 a.e. ===}J[?d[ IPTn,pfldP = [ IPTn(J·1 @p)ld(m x m[?d) ---+ 0JXX[?d q, JXX[?d ¢as n ---+ 00; equivalently (taking F(x, y) := f(x, y)p(y)),[ F(x,y)dy=Oa.e. ===} [ IPTnFld(mxm[?d)---+OJ[?d JXX[?d ¢as n ---+ 00.To prove (*), we first claim that~1 for A > 1, h E L1(m) and f E L1(]Rd),as k ---+ 00 where C = 2~m(B(O, 1)) and ~~Z ---+ 0 a.e..PROOF As can be checked,Denoting E(H) := Ix Hdm for H E L1(m), we haveBy the Cauchy-Schwartz inequality,(3)whereas8 JON AARONSON AND MANFRED DENKER1 :s: 1 IE(PTn,.(h(·)f(y - <Pnk(·))l[1c1>nk(o))I:5,\n kj)ldylyl>2,\nk lyl>2,\nk+1 IE(PTnk (h(·)f(y - <Pnk (.))l[1c1>nkOI>,\nkj))ldy = I + II.lyl>2,\nkHere as k -+ 00:(4). cl>nk 0 . dSInce ,\nk -+ a.e., an(5)I:S: 1 E(lhllf(y - <Pnk)11[1c1>n,.(-)I:5,\nkj)dylyl>2,\nk= E(lhI1[1c1>nk,:5,\nkjl If(y - <Pnk)ldY)lyl>2,\nk:s: E(lhl)1 If(y)ldy -+ 0,Iyl>,\nkSubstituting (3),(4) and (5) into (2) proves'1. 0To complete the proof of (*), let FE LI(m x m]Rd) satisfy f]Rd F(x,y)dy = 0 form-a.e. x E X and fix E > o. We show thatStandard approximation techniques show that V E > 0, :J N E N, hI, .. " hN EL, 9I, ... ,9N E LI(JRd) such that f]Rd9k(y)dy = 0 (1:S: k:S: N) andNIIF - L hk (9 9kll£l(rnXrnRk) < ~.k=INext, it follows from theorems 1.6.3 and 1.6.4 in [Rud] that:J !I, ... ,fN E L I nL2 such that• [ik =1= 0] is compact and bounded away from 0 (1:S: k :s: N);and• Ilfk - 9kll£l(rnR d) < 2NllhkII L1 (Tn) (1:S: k :s: N), whenceN N NII L hk (9 fk - L hk (9 9kll£l(rnXrnad) :s: L Ilhkll£l(rn) . Ilfk - 9kll£l(]Rd) < ~,k=I k=I k=INIIF - L hk (9 fkll£l(rnXrnad) < Ek=ION EXACT GROUP EXTENSIONS 9where h ELand f E L 1 n L 2 is such that [j =1= 0] is compact and bounded awayfrom O.We claim'2 If h ELand f E L 1 n L2 is such that [j =1= 0] is compact and bounded awayfrom 0, then :3 0 0, 0 (h 0 f)(x, y) is 'Y .-j('Y)P;h(x) and Plancherel's formula, we haveproving '2. 0To finish the proof of theorem 2, we claim'3 if (6) holds for hE Land f E L 1 n L 2 , then(7)PROOFFix.A> 1 such that .A~p < 1. Suppose that ~:Z -+ 0 a.e.. Using (6), we haveby'l,as k -+ 00; establishing (7) since IIPT;(h 0 f)111 1. 0This completes the proof of theorem 2.References[A][A-Dl][A-D-U]J. Aaronson, An introduction to infinite ergodic theory, Mathematical surveys andmonographs 50, American Mathematical Society, Providence, R.I, U.S., 1997.J. Aaronson, M. Denker, Local limit theorems for Gibbs-Markov maps, Preprint(1996).J. Aaronson, M. Denker, M. Urbanski, Ergodic theory for Markov fibred systemsand parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.10[Ad][Bo][Bo-Ru][D-L][D-F][G][IT-M][L][Mor][Nag][Rud][Rou][Ry][S]JON AARONSON AND MANFRED DENKERR. Adler, F -expansions revisited, Recent advances in topological dynamics, LectureNotes in Math., vol. 318, Springer, Berlin, Heidelberg, New York, 1973, pp. 1-5.R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-460.R. Bowen, D. Ruelle, The ergodic theory of axiom A flows, Inventiones math. 29(1975), 181-202.Y. Derriennic, Y, M. Lin, Sur Ie comportement asymptotique des puissances deconvolution d'une probabilite, Ann. Inst. H. Poincari Probab. Statist. 20 (1984),127-132.W. Doeblin, R. Fortet, Sur des chaines a liaison completes, Bull. Soc. Math. deFrance 65 (1937), 132-148.Y. Guivarc'h, Proprietes ergodiques, en mesure infinie, de certains systemes dy-namiques fibres, Ergod. Th. and Dynam. Sys. 9 (1989), 433-453.lonescu-Tulcea, G. Marinescu, Theorie ergodique pour des classes d'operations noncomplet-ement continues, Ann. Math. 47 (1950), 140-147.M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Ge-biete 19 (1971), 231-242.T. Morita, Local limit theorems and density of periodic points for Lasota- Yorketransformations, J. Math.Soc. Japan 46 (309-343).S.V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab.Appl. 2 (1957), 378-406.W. Rudin, Fourier analysis on groups, Wiley-Interscience, New York, 1962.J. Rousseau-Egele, Un theorem de la limite locale pour une classe de transformationsmonotones et dilatantes par marceaux, Ann. Probab. 11 (1983), 722-788.M. Rychlik, Bounded variation and invariant measures, Stud. Math. 76 (1983),69-80.K. Schmidt, Cocycles of Ergodic Transformation Groups, Lect. Notes in Math. Vol.1, Mac Millan Co. of India, 1977.AARONSON: SCHOOL OF MATHEMATICAL SCIENCES, TEL AVIV UNIVERSITY, 69978 TEL AVIV,ISRAEL.E-mail address:aaro<omath.tau.ac.ilDENKER: INSTITUT FUR MATHEMATISCHE STOCHASTIK, UNIVERSITAT GOTTINGEN, LOTZE-STR. 13, 37083 GOTTINGEN, GERMANYE-mail address:denker<Omath.uni-goettingen.de

On exact group extensions

Aaronson, J

Denker, M

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On exact group extensionsCitation for published version (APA):Aaronson, J., & Denker, M. (1999). On exact group extensions. (Report Eurandom; Vol. 99047). Eurandom.Document status and date:Published: 01/01/1999Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Report 99-047On Exact Group ExtensionsJon AaronsonManfred DenkerISSN 1389-2355ON EXACT GROUP EXTENSIONSJON AARONSON AND MANFRED DENKERABSTRACT. We give conditions for the exactness of JR.d-extensions.§O INTRODUCTIONA nonsingular transformation (X, B, m, T) of a standard probability space iscalled a fibred system if there is a generating measurable partition 0 such thatT : a ~ Ta is invertible, nonsingular for a E 0, and a Markov map (or Markovfibred system) if in addition, Ta E 0'(0) mod m \:j a E o.Write 0 = {as : s E S} and endow SN with its canonical (Polish) producttopology. Letn~ = {s = (81,82,.") E sN: m(n T-ka Sk ) > 0 \:j n:;::: I},k=1then ~ is a closed, shift invariant subset of SN, and there is a measurable map4> : ~ ~ X defined by {4>(st, S2, ..• )} := n~=1T-(k-l)a Sk 'The closed support of the probability m' = m 0 4>-1 is ~, and 4> is a conjugacyof (X,B,m,T) with (~,B(~),m',shift). Thus we may, and sometimes do, assumethat X = ~, T is the shift, and 0 = {[s] : s E S}.For n :;::: 1, there are m-nonsingular inverse branches of T denotedVa : Tn a ~ a and with Radon Nikodym derivatives denoted, dm 0 VaVa := dm .Let (X, B, m, R) be a nonsingular transformation of a standard probability space.The Frobenius-Perron operators PRn = PRn,m : L 1(m) ~ L 1(m) are defined byand for the locally invertible (X, B, m, T, 0) (as above) have the formPTnf = L ITn a V:' f 0 Va'aEQ~-l1991 Mathematics Subject Classification. Primary: 28D05, 60B15; Secondary: 58F15, 58F19,58F30.©July 1999, revision 4/10/99Typeset by A,;\IfS-'lEX12 JON AARONSON AND MANFRED DENKERA locally invertible map (X, B, m, T, a) has:the Renyi property if:J C> 1 such that V n ~ 1, a E a~-l, m(a) > 0:I~!~:~ I ::; C for m x m-a.e. (x, y) E Tna x Tn a.It is well known (a proof is recalled in [A-D-U]) that any topologically mixingprobability preserving Markov map with the Renyi property is exact in the sensethat nn>1 T-n B = {0, X} mod m.Examples include:• topological Markov shifts equipped with Gibbs measures ([Bo],[Bo-Ru]) and• uniformly expanding, piecewise onto C 2 interval maps T : [0, 1] ~ [0,1] satisfyingAdler's condition sUPxE[O,1] Ir~~)~1 < 00 ([Ad]);or, more generally,• Gibbs-Markov maps as in [A-D1].Now let ¢ : X ~ ]Rd be measurable and consider the skew product Tlj> : X x]Rd ~X x]Rd defined by Tlj>(x, y) := (Tx, y+¢(x)) with respect to the (invariant) productmeasure m x mRd where mRd denotes Lebesgue measure.We say that ¢ is aperiodic if ,(¢) = zhh 0 T has no nontrivial solution in , EJRd , Z E S1 and h : X ~ S1 measurable. It is not hard to show that if Tlj> is ergodic,and T is weakly mixing, then Tlj> is weakly mixing iff ¢ is aperiodic.We're interested in the exactness of Tlj>.We establish two (partial) results in this direction.Theorem 1.Suppose that (X, B, m, T, a) is a probability preserving Markov map with theRenyi property. Let N ~ 1 and ¢ : X ~ ]Rd be a~-1-measurable (i.e. ¢(x) =¢(a~-1(x)) where x E a~-1(x) E a~-1).If Tlj> is topologically mixing, then Tlj> is exact.For the other result, we assume that (X, B, m, T, a) is an exact probabilitypreserving locally invertible map with the property that for some Banach space(L, II· IlL) of functions with II ·112 ::; II . IlL, such that PT : L ~ Land :JM > 0, () E(0,1) such thatIIPTnf - i fdmllL ::; M(}nllfilL V f E L.This property can be obtained as a consequence of the quasi compactness of Doeblin-Fortet operators, see [D-F]' [IT-M]).Given ¢ : X ~ ]Rd measurable, we define the characteristic function operatorsPt(J) = PT(ei(t,lj» J) (t E JRd).We assume also that Pt : L ~ L (t E ]Rd) and that t 1-+ Pt is continuous(JRd ~ Hom(L, L).It is shown in [Nag] (see also theorem 4.1 of [A-Dl]) that(i) there are constants € > 0, K > °and () E (0,1); and continuous functions,X : B(O, €) ~ Bc(O, 1), g : B(O, €) ~ L such thatIlptnh - 'x(t)ng(t) Ix hdmllL ::; K(}nllhil L Viti < €, n ~ 1, hE L;andON EXACT GROUP EXTENSIONS(ii) in case ¢> is aperiodic, then '\f 0 < 0 < M < 00, :J K > 0, 0 : X _ ]Rd uniformlyHolder continuous on partition sets. Here L is a space of Holder continuous func-tions f : X-C.• (see [Rou], [Ry]), X = [0,1], m Lebesgue measure, Q a partition of X mod minto open intervals, and T : a - Ta an invertible, m-nonsingular homeomorphismfor each a E Q with inf IT'I > 1 and i, of bounded variation on X; and ¢> : X _ ]Rdeither: of bounded variation on X; or constant on each a E Q.Set ¢>n = ¢> + ¢> 0 T + ... + ¢> 0 Tn-I.Theorem 2.Suppose that(0) '\f ,X > 1 :J nk - 00 such that ¢>nk _ 0 a.e. as k - 00,Xnkand that ¢> is aperiodic;then Tel> is exact.Remarks.1) Theorem 2 generalises the corresponding theorem on page 443 in [G].2) The condition (0) is satisfied if m-dist (¢» is in the domain of attraction of astable law.3) The condition (0) is not satisfied iff :J ,X > 1 and c > 0 such that m([!¢>n I >,Xn]) ~ c '\f n ~ 1 and there are independent processes like this.§1 FROBENIUS-PERRON OPERATORS, EXACTNESS AND RELATIVE EXACTNESSLet (X, B, m, R) be a nonsingular transformation of a standard probability space.The tail (J-algebra of (X, B, m, R) is T(R) := n:=l R-nB and the nonsingulartransformation R is called exact if = {0, X} mod m.Theorem 1.1 [D-L].IIPRnflh -IIE(fIT(R))III as n - 00 '\f f E L1(m).In particular (see [L]), R is exact iff IIPRnflh - O'\f f E L1(m), Ix fdm = O.Proof.First note that IPT fl :::; PT!fl whence IIPRnfll1 1and:J limn _ oo IIPRnflll' Next,'\f n ~ 1 :J gn E LOO(B) with Ix(PRnf)gndm = IIPRnflll' whenceIIPRnfll1 = Lfgn 0 Rndm.By weak * compactness, :J nk - 00 and 9 E LOO(B) such that gnk 0 Rnk ---" 9 weak* in £OO(B).4 JON AARONSON AND MANFRED DENKERIt follows that g E Loo(T(R)), IIglloo ~ 1 and limn_>oo IIPRnfl11 = Ix fgdm.Thuslim IIPRnflh ~ sup { r fhdm: hE Loo(T(R)), Ilhll oo ~ I} = IIE(fIT(R))111'n->oo JXTo show the converse inequality, note that :3 g E Loo(T(R)), IIglloo = 1 suchthatIIE(fIT(R))lh = i E(fIT(R))gdm = i fgdmwhence V n ~ 1, :3 gn E Loo(B), g = gn 0 Rn andIIE(fIT(R))lh = i fgdm = i fgn 0 Rndm = i (PRnf)gndm ~ IIPRnflh·oLet (X, B, m, R) and (Y, C, jJ, S) be nonsingular transformations of standardprobability spaces. A factor map is a function 7r : X ----+ Y satisfying 7r-1C CB, 7r 0 T = So 7r, m 0 7r- 1 = jJ.The fibre expectation of the factor map 7r : X ----+ Y is an operatorf ~ E(fI7r), L1(X,B,m) ----+ L1(Y,C,jJ)defined by Iy E(fI7r)gdjJ = Ix fg 0 7rdm.The factor map 7r : X ----+ Y is called relatively exact iff E L1(B), E(fI7r) = 0 a.e. ===? IIPRnflh ----+ O.The corollary below appears in [G]. For the convenience of the reader, we supplya (possibly different) proof.Proposition 1.2. Suppose that 7r : X ----+ Y is relatively exact, then T(R) =7r- 1T(S) mod m.Proof.Evidently, 7r- 1T(S) ~ T(R). We show that 7r-1T(S) 2 T(R).By relative exactness and theorem 1.1, if f E L1(B) and E(f/7r) = 0 a.e., thenIx fgdm = 0 V g E Loo(T(R)).Thus if f E L2 (B) e L2 (7r- 1C), then E(fI7r) = 0 a.e. and soi fgdm = 0 V g E Loo(T(R)), ===? f..l L2 (T(R)).Thus L2 (B) e L2 (7r- 1C) c L2(B) e L2(T(R)) whence L2 (T(R)) C L2 (7r- 1C) andT(R) C 7r-1C mod m.To see that in fact T(R) ~ 7r-1T(S) mod m, fix N ~ 1, thenT(R) = nR-nB = n R-nBn~1 n~N+1= R-NT(R) C R-N7r-1C = 7r- 1S-NC.Taking the intersection over N shows the claim. 0Corollary 1.3 ([G], proposition 1).If S is exact and 7r : X ----+ Y is relatively exact, then T is exact.ON EXACT GROUP EXTENSIONS§2 PROOF OF THEOREM 1For a nonsingular transformation (X, B, m, R), define the tail relation of R:5Evidently 'r(R) is an equivalence relation and if (X, B, m) is standard, then'r(R) E B(X x X).If R is locally invertible, then 'r(R) has countable equivalence classes and isnonsingular in the sense that m('r(R)(A)) = °\f A E B, m(A) = °where'r(R)(A) := {y EX: :3 x E A (x, y) E 'r(R)}.A set A E B(X) is invariant under the equivalence relation 'r E B(X x X) if'r(A) = A and the equivalence relation 'r is called ergodic if'r-invariant sets haveeither zero, or full measure.The collection of invariant sets under 'r(R) is the tail a-algebra T(R) (whencethe name" tail relation").In order to prove theorem 1, it suffices to show that 'r(T<t» is ergodic.The tail relation of T<t> is given by'r(T<t> )= {((x, s), (y, t)) E (X X G)2: :3 n 2: 0, Tnx = Tny, S - t = cPn(Y) - cPn(x)}= {((x, S), (y, t)) E (X X G)2: (x, y) E'r(T), ¢(x, y) = S - t}where ¢ : 'r(T) -+ lRd is defined by ¢(x, y) := 'L':=o(cP(Tn y) - cP(Tn x )).We prove that 'r(T<t» is ergodic by the method of Schmidt (explained in [S]), byshowing that \f t E lRd , U a neighbourhood of t and A E B m(A) > 0, :3 B EBB cA and T : B -+ B nonsingular such that (x, T(X)) E 'r(T) and ¢(x, T(X)) E U \f x EB.This boils down to showing that\f A E B+ go E lRd TJ > 0, :3 B E B+ B c A, n 2: 1and T : B -+ T B c A nonsingular such that(+) Tn 0 T Tn and II cPn 0 T - cPn - go II < TJ on B.The proof of (+) will be written as a sequence of minor claims, '0,'1, ....'0 We first claim that there is no loss in generality in assuming that N = 1 (i.e.that cP : X -+ lRd is a-measurable). This is because (X, B, m, T, a~-l) is also aprobability preserving Markov map with the Renyi property and inducing the same(shift) topology on X as (X, B, m, T, a).'1 \f s, t E S, :3 '" = "'s,t 2: 1 and a = as,t = [a1, ... aK]' b = bs,t = [bI, ... bK] Ea~-l, a1 = b1 = S aK= bK= t such that IlcPK(b) - cPK(a) - goll < TJ.This follows from topological mixing of T<t>.By the Renyi property, :3 M > 1 such that6 JON AARONSON AND MANFRED DENKERGiven U = [Ub ... ' un] E Q~-1 with Un = t, define T = Tu : unT-na -+ unT-nbby'2 T = Tu : un T-n a -+ un T-nb is invertible nonsingular and d;;';T = M±4:~~~.PROOF1 dmOTd dm = m(unT-nbnc)unT-nanc m= M±2 m(b) m(u)m(b)m(c)m(a)= M±4 m(b) m(u n T-na n c).m(a)o'3 PROOF OF :j:Fix 0 < E < M-1 min{m(as ,d, m(bs,t)}, thenLet 8 > 0 be so small that 8 < m~~~(a~).:3 n ~ 1 and u E Q~-1 such that m(A n u) ~ (1- 8)m(u) and [8] e T[un].Consider Tu : un T-n a -+ un T-nb as in '2. Evidently Tn+K 0 T Tn+K andII<Pn+K 0 T - <Pn+K - goll < 'TJ on un T-na.To complete the proof we claim that :3 B E B+ B e An u n T-n a such thatTBeA.To see this, note thatm(u n T-na n A) ~ m(u n T-na) - m(u \ A) ~ (E - 8)m(u),whence using '2,m(b) m(b)(E - 8)m(T(u n T-na n A)) ~ M4m(a) m(u n T-na n A) ~ M4m (a) m(u).Since T(U n T-na n A) e u, the condition on 8 > 0 ensures that m(T(u n T-n a nA) n A) > 0 whence m(B) > 0 where B := T- 1 (T(U n T-n a n A) n A) eA. 0§3 PROOF OF THEOREM 2We prove theorem 2 via corollary 1.3. To do this, we must consider Ttl> as anonsingular transformation with respect to some probability P rv m x mJRd.Let p : JR.d -+ ~ be continuous with JJRd p(y)dy = 1 and define a probabilityP on X x JR.d by dP(x, y) := p(y)dm(x)dy; then (X x JR.d, B(X X JR.d), P, Ttl» is anonsingular transformation with Frobenius-Perron operators given byON EXACT GROUP EXTENSIONS1PTn p f (x, y) = -() PTn (J . 1@p)(x,y)q,' py q,7where PTn := PTn mXm d·q, q,' IRConsider the map 7r : X x ]Rd ---+ X defined by 7r(x, y) = x. This is a factor mapas it satisfies 7r- 1B(X) C B(X x ]Rd), 7r 0 T</> = To 7r, Po 7r-1 = m.The fibre expectation of 7r is given byE(JI7r)(x) = [ f(x, y)p(y)dy (J E L1(X X ]Rd, B(X X ]Rd), P)).J[?dBy corollary 1.3 and exactness of T, it suffices to show that 7r is relatively exact.To do this, we show that[ f(x, y)p(y)dy = 0 a.e. ===}J[?d[ IPTn,pfldP = [ IPTn(J·1 @p)ld(m x m[?d) ---+ 0JXX[?d q, JXX[?d ¢as n ---+ 00; equivalently (taking F(x, y) := f(x, y)p(y)),[ F(x,y)dy=Oa.e. ===} [ IPTnFld(mxm[?d)---+OJ[?d JXX[?d ¢as n ---+ 00.To prove (*), we first claim that~1 for A > 1, h E L1(m) and f E L1(]Rd),as k ---+ 00 where C = 2~m(B(O, 1)) and ~~Z ---+ 0 a.e..PROOF As can be checked,Denoting E(H) := Ix Hdm for H E L1(m), we haveBy the Cauchy-Schwartz inequality,(3)whereas8 JON AARONSON AND MANFRED DENKER1 :s: 1 IE(PTn,.(h(·)f(y - <Pnk(·))l[1c1>nk(o))I:5,\n kj)ldylyl>2,\nk lyl>2,\nk+1 IE(PTnk (h(·)f(y - <Pnk (.))l[1c1>nkOI>,\nkj))ldy = I + II.lyl>2,\nkHere as k -+ 00:(4). cl>nk 0 . dSInce ,\nk -+ a.e., an(5)I:S: 1 E(lhllf(y - <Pnk)11[1c1>n,.(-)I:5,\nkj)dylyl>2,\nk= E(lhI1[1c1>nk,:5,\nkjl If(y - <Pnk)ldY)lyl>2,\nk:s: E(lhl)1 If(y)ldy -+ 0,Iyl>,\nkSubstituting (3),(4) and (5) into (2) proves'1. 0To complete the proof of (*), let FE LI(m x m]Rd) satisfy f]Rd F(x,y)dy = 0 form-a.e. x E X and fix E > o. We show thatStandard approximation techniques show that V E > 0, :J N E N, hI, .. " hN EL, 9I, ... ,9N E LI(JRd) such that f]Rd9k(y)dy = 0 (1:S: k:S: N) andNIIF - L hk (9 9kll£l(rnXrnRk) < ~.k=INext, it follows from theorems 1.6.3 and 1.6.4 in [Rud] that:J !I, ... ,fN E L I nL2 such that• [ik =1= 0] is compact and bounded away from 0 (1:S: k :s: N);and• Ilfk - 9kll£l(rnR d) < 2NllhkII L1 (Tn) (1:S: k :s: N), whenceN N NII L hk (9 fk - L hk (9 9kll£l(rnXrnad) :s: L Ilhkll£l(rn) . Ilfk - 9kll£l(]Rd) < ~,k=I k=I k=INIIF - L hk (9 fkll£l(rnXrnad) < Ek=ION EXACT GROUP EXTENSIONS 9where h ELand f E L 1 n L 2 is such that [j =1= 0] is compact and bounded awayfrom O.We claim'2 If h ELand f E L 1 n L2 is such that [j =1= 0] is compact and bounded awayfrom 0, then :3 0 0, 0 (h 0 f)(x, y) is 'Y .-j('Y)P;h(x) and Plancherel's formula, we haveproving '2. 0To finish the proof of theorem 2, we claim'3 if (6) holds for hE Land f E L 1 n L 2 , then(7)PROOFFix.A> 1 such that .A~p < 1. Suppose that ~:Z -+ 0 a.e.. Using (6), we haveby'l,as k -+ 00; establishing (7) since IIPT;(h 0 f)111 1. 0This completes the proof of theorem 2.References[A][A-Dl][A-D-U]J. Aaronson, An introduction to infinite ergodic theory, Mathematical surveys andmonographs 50, American Mathematical Society, Providence, R.I, U.S., 1997.J. Aaronson, M. Denker, Local limit theorems for Gibbs-Markov maps, Preprint(1996).J. Aaronson, M. Denker, M. Urbanski, Ergodic theory for Markov fibred systemsand parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.10[Ad][Bo][Bo-Ru][D-L][D-F][G][IT-M][L][Mor][Nag][Rud][Rou][Ry][S]JON AARONSON AND MANFRED DENKERR. Adler, F -expansions revisited, Recent advances in topological dynamics, LectureNotes in Math., vol. 318, Springer, Berlin, Heidelberg, New York, 1973, pp. 1-5.R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-460.R. Bowen, D. Ruelle, The ergodic theory of axiom A flows, Inventiones math. 29(1975), 181-202.Y. Derriennic, Y, M. Lin, Sur Ie comportement asymptotique des puissances deconvolution d'une probabilite, Ann. Inst. H. Poincari Probab. Statist. 20 (1984),127-132.W. Doeblin, R. Fortet, Sur des chaines a liaison completes, Bull. Soc. Math. deFrance 65 (1937), 132-148.Y. Guivarc'h, Proprietes ergodiques, en mesure infinie, de certains systemes dy-namiques fibres, Ergod. Th. and Dynam. Sys. 9 (1989), 433-453.lonescu-Tulcea, G. Marinescu, Theorie ergodique pour des classes d'operations noncomplet-ement continues, Ann. Math. 47 (1950), 140-147.M. Lin, Mixing for Markov operators, Z. Wahrscheinlichkeitstheorie und Verw. Ge-biete 19 (1971), 231-242.T. Morita, Local limit theorems and density of periodic points for Lasota- Yorketransformations, J. Math.Soc. Japan 46 (309-343).S.V. Nagaev, Some limit theorems for stationary Markov chains, Theor. Probab.Appl. 2 (1957), 378-406.W. Rudin, Fourier analysis on groups, Wiley-Interscience, New York, 1962.J. Rousseau-Egele, Un theorem de la limite locale pour une classe de transformationsmonotones et dilatantes par marceaux, Ann. Probab. 11 (1983), 722-788.M. Rychlik, Bounded variation and invariant measures, Stud. Math. 76 (1983),69-80.K. Schmidt, Cocycles of Ergodic Transformation Groups, Lect. Notes in Math. Vol.1, Mac Millan Co. of India, 1977.AARONSON: SCHOOL OF MATHEMATICAL SCIENCES, TEL AVIV UNIVERSITY, 69978 TEL AVIV,ISRAEL.E-mail address:aaro<omath.tau.ac.ilDENKER: INSTITUT FUR MATHEMATISCHE STOCHASTIK, UNIVERSITAT GOTTINGEN, LOTZE-STR. 13, 37083 GOTTINGEN, GERMANYE-mail address:denker<Omath.uni-goettingen.de

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