Decay of correlations for non-H¨older observables 4 0 0 Vincent Lynch 2 n February 1, 2008 a J 0 3 Abstract We consider the general question of estimating decay of correlations ] S for non-uniformly expanding maps, for classes of observables which are D much larger than the usual class of H¨older continuous functions. Our results give new estimates for many non-uniformly expanding systems, . h including Manneville-Pomeau maps, many one-dimensional systems with t a critical points, and Viana maps. In many situations, we also obtain a m Central LimitTheorem foramuchlargerclass of observables thanusual. Our main tool is an extension of the coupling method introduced by [ L.-S. Young for estimating rates of mixing on certain non-uniformly ex- 1 panding tower maps. v 2 3 1 Introduction 4 1 In this paper, we are interested in mixing properties (in particular, decay of 0 correlations) of non-uniformly expanding maps. Much progress has been made 4 0 in recent years, with upper estimates being obtained for many examples of / such systems. Almost invariably, these estimates are for observables which are h t Ho¨lder continuous. Our aim here is to extend the study to much larger classes a of observables. m Let f :(X,ν)(cid:9) be some mixing system. We define a correlation function : v Xi n(ϕ,ψ;ν)= (ϕ fn)ψdν ϕdν ψdν C ◦ − (cid:12)Z Z Z (cid:12) r (cid:12) (cid:12) a for ϕ,ψ 2. The rate at w(cid:12)(cid:12)hich this sequence decays to z(cid:12)(cid:12)ero is a measure ∈ L of how quickly ϕ fn becomes independent from ψ. It is well known that ◦ for any non-trivial mixing system, there exist ϕ,ψ 2 for which correlations ∈ L decay arbitrarily slowly. For this reason, we must restrict at least one of the observables to some smaller class of functions, in order to get an upper bound for . n C Here, we present a result which is general in the context of towers, as in- troduced by L.-S. Young ([Yo]). There are many examples of systems which admit such towers, and we shall see that under a fairly weak assumption on the relationship between the tower and the system (which is satisfied in all the 1 examples we mention) we get estimates for certain classes of observables with respect to the system itself. One of the main strengths of this method is that these classes of observables may be defined purely in terms of their regularity with respect to the manifold; this contrasts with some results, where regularity is considered with respect to some Markov partition. Acknowledgements. This paper is based on work contained in my Ph.D. thesis, written at the University of Warwick. Many thanks to my supervisor, Stefano Luzzatto, and also to Omri Sarig and Peter Walters. I am grateful for many conversationswith Mark Holland, Frederic Paccaut, Gavin Band and Mike Todd. 2 Statement of results Let us start by defining the classes of observables we consider. Let (X,d) be a metric space. For a given function ψ :X R, for each ε>0 we write → (ψ):=sup ψ(x) ψ(y) :d(x,y) ε . ε R {| − | ≤ } We see that (ψ) 0 as ε 0 if and only if ψ is uniformly continuous. We ε R → → define classes of functions corresponding to different rates of decay for (ψ), ε R as follows: Class (R1,γ), γ (0,1]: ψ (R1,γ) if (ψ)= (εγ). ε ∈ ∈ R O Class (R2,γ), γ (0,1): ψ (R2,γ) if (ψ)= (exp logεγ ). ε ∈ ∈ R O {−| | } Class (R3,γ), γ >1: ψ (R3,γ) if (ψ)= (exp (log logε)γ ). ε ∈ R O {− | | } Class (R4,γ), γ >1: ψ (R4,γ) if (ψ)= ( logε−γ). ε ∈ R O | | We write (Rn) = (Rn,γ) for each n. Note (R1) is the class of functions γ ∪ which are Ho¨lder continuous. Also note that (R1) (R2) (R3) (R4) with ⊂ ⊂ ⊂ the inclusions strict, and in fact (Rn,γ) (R(n+1)) for everyn=1,2,3,with ⊃ γ in the appropriate interval. It is not difficult to find observables with exactly any of these regularities; for instance, the function ψ :[0,1] R given by 2 → logx−γ x>0 ψ(x)= | | 0 x=0 (cid:26) for some γ >1, has the property that (ψ)=ψ(ε) for all small ε. ε R Beforewestate the exacttechnicalresult,weshallillustrate itsimplications by stating results for a number of different classes of system. In each case, the estimate given for Ho¨lder continuous observables (i.e. case (R1)) has been obtained previously, but we are able to give estimates for observables in each of the other classes, most of which are new. Often the same estimates given in the Ho¨lder case also apply to one of the larger classes; it seems that when the rate of mixing for Ho¨lder observables is sub-exponential, we can expect the samerateofmixingtoholdforalargerclassofobservables. Forinstance,when Ho¨lder observables give polynomial mixing, we tend to get the same speed of mixing for observables in some class (R4,γ), γ >1. 2 All of our results shall take the following form. Given a system f : X (cid:9), a mixing acip ν, and ϕ ∞(X,ν), ψ , for some class = (Ri,γ) as above, ∈ L ∈ I I we obtain in each example an estimate of the form (ϕ,ψ;ν) ϕ C(ψ)u , n ∞ n C ≤k k where is the usual norm on ∞(X,ν), C(ψ) is a constant depending ∞ k ·k L on f and ψ, and (u ) is some sequence decaying to zero with rate determined n by f and (ψ). Notice that we make no assumption on the regularity of the ε R observableϕ;whendiscussingtheregularityclassofobservables,weshallalways bereferringtothechoiceofthefunctionψ. (Thisisnotatypical,althoughsome existing results do require that both functions have some minimum regularity.) For brevity,weshallsimply giveanestimate for u in the statementofeach n result. For each example we also have a Central Limit Theorem for those ob- servableswhichgivesummable decayofcorrelations,andarenotcoboundaries. We recall that a real-valued observable ψ satisfies the Central Limit Theorem for f if there exists σ >0 such that for every interval J R, ⊂ n−1 ν x X : 1 ϕ(fj(x)) ϕdν J 1 e−2tσ22dt.  ∈ √n − ∈ → σ√2π  Xj=0(cid:18) Z (cid:19)  ZJ Note that the range of examples givenin the following subsections is meant to be illustrative rather than exhaustive, and so we shall miss out some simple generalisations for which essentially the same results hold. We shall instead try to make clear the conditions needed to apply these results, and direct the reader to the papers mentioned below for further examples which satisfy these conditions. 2.1 Uniformly expanding maps Let f : M (cid:9) be a C2-diffeomorphism of a compact Riemannian manifold. We say f is uniformly expanding if there exists λ > 1 such that Df v λ v x k k ≥ k k for all x M, and all tangent vectors v. Such a map admits an absolutely ∈ continuous invariant probability measure µ, which is unique and mixing. Theorem 1 Let ϕ ∞(M,µ), and let ψ : M R be continuous. Upper ∈ L → bounds are given for (u ) as follows: n if ψ (R1), then u = (θn) for some θ (0,1); n • ∈ O ∈ if ψ (R2,γ), for some γ (0,1), then u = (e−nγ′) for every γ′ <γ; n • ∈ ∈ O ′ if ψ (R3,γ), for some γ >1, then u = (e−(logn)γ ) for every γ′ <γ; n • ∈ O for any constant C > 0 there exists ζ < 1 such that if ψ (R4,γ) for ∞ • ∈ some γ >ζ−1, and (ψ)<C , then u = (n1−ζγ). ∞ ∞ n R O 3 Furthermore, the Central Limit Theorem holds when ψ (R4,γ) for suffi- ∈ ciently large γ, depending on (ψ). ∞ R Such maps are generally regardedas being well understood, and in particu- lar, results of exponential decay of correlations for observables in (R1) go back totheseventies,andtheworkofSinai,RuelleandBowen([Si],[R],[Bo]). Fora moremodernperspective,seeforinstancethe booksofBaladi([Ba])andViana ([V2]). I have not seen explicit claims of similar results for observables in classes (R2 4). However, it is well known that any such map can be coded by a − one-sided full shift on finitely many symbols, so an analogous result on shift spaces would be sufficient, and may well already exist. The estimates here are probably not sharp, particularly in the (R4) case. Theotherexamplesweconsiderarenotingeneralreducibletofinitealphabet shift maps, so we can be more confident that the next set of results are new. 2.2 Maps with indifferent fixed points These are perhaps the simplest examples of strictly non-uniformly expand- ing systems. Purely for simplicity, we restrict to the well known case of the Manneville-Pomeaumap. Theorem 2 Let f : [0,1] (cid:9) be the map f(x) = x+x1+α (mod 1), for some α (0,1), and let ν be the unique acip for this system. For ϕ,ψ : [0,1] R ∈ → with ϕ bounded and ψ continuous, for every constant C >0 there exists ζ <1 ∞ such that if ψ (R4,γ) for some γ >2ζ−1, with (ψ)<C , then ∞ ∞ ∈ R if γ =ζ−1(τ +1), then u = (n1−τ logn); n • O otherwise, u = (max(n1−τ,n2−ζγ)); n • O where τ =α−1. In particular, when γ > 3 the Central Limit Theorem holds. ζ In the case where ψ (R4,γ) for every large γ, this gives un = (n1−α1), ∈ O which is the bound obtained in [Yo] for ψ (R1). We do not give separate ∈ estimatesforobservablesinclasses(R2)and(R3),asweobtainthesameupper boundineachcase. Notethatthepolynomialupperboundfor(R1)observables is known to be sharp ([Hu]), and hence the above gives a sharp bound in the (R2) and (R3) cases, and for (R4,γ) when γ is large. The above results apply in the more general 1-dimensional case considered in [Yo], where in particular a finite number of expanding branches are allowed, and it is assumed that xf′′(x) xα near the indifferent fixed point. ≈ In our remaining examples, estimates will invariably correspond to either the above form, or that of Theorem 1, and we shall simply say which is the case, specifying the parameter τ as appropriate. 4 2.3 One-dimensional maps with critical points Let us consider the systems of [BLS]. These are one-dimensional multimodal maps, where there is some long-term growth of derivative along the critical orbits. Let f : I I be a C3 interval or circle map with a finite critical set → C and no stable or neutral periodic orbit. We assume all critical points have the same critical order l (1, ); this means that for each c , there is some ∈ ∞ ∈ C neighbourhood in which f can be written in the form f(x)= ϕ(x c)l+f(c) ±| − | for some diffeomorphism ϕ : R R fixing 0, with the allowed to depend on → ± the sign of x c. − For c , let D (c) = (fn)′(f(c)). From [BLS] we know there exists an n ∈ C | | acip µ provided D−2l1−1(c)< c . n ∞ ∀ ∈C n X If f is not renormalisable on the support of µ then µ is mixing. Theorem 3 Let ϕ ∞(I,µ), and let ψ be continuous. ∈L Case 1: Suppose there exist C > 0, λ > 1 such that D (c) Cλn for all n ≥ n 1, c . Then we have estimates for (u ) exactly as in the uniformly n ≥ ∈ C expanding case (Theorem 1). Case 2: Suppose there exist C > 0, α > 2l 1 such that D (c) Cnα for n − ≥ all n 1, c . Then we have estimates for (u ) as in the indifferent fixed n point ≥case (T∈heoCrem 2) for every τ < α−1. l−1 In particular, the Central Limit Theorem holds in either case when ψ ∈ (R4,γ) for sufficiently large γ, depending on (ψ). ∞ R Again, we have restricted our attention to some particular cases; analogous results should be possible for the intermediate cases considered in [BLS]. In particular, for the class of Fibonacci maps with quadratic critical points (see [LM]) we obtain estimates as in Theorem 2 for every τ >1. 2.4 Viana maps Next we consider the class of Viana maps, introduced in [V1]. These are ex- amples of non-uniformly expanding maps in more than one dimension, with sub-exponential decay of correlations for Ho¨lder observables. They are notable for being possibly the first examples of non-uniformly expanding systems in more than one dimension which admit an acip, and also because the attractor, andmanyofitsstatisticalproperties,persistinaC3 neighbourhoodofsystems. Let a be some real number in (1,2) for which x=0 is pre-periodic for the 0 system x a x2. We define a skew product fˆ:S1 R(cid:9) by 0 7→ − × fˆ(s,x)=(ds mod 1,a +αsin(2πs) x2), 0 − 5 wheredisaninteger 16,andα>0isaconstant. Whenαissufficientlysmall, ≥ thereisacompactintervalI ( 2,2)forwhichS1 I ismappedstrictlyinside ⊂ − × its own interior, and fˆadmits a unique acip, which is mixing for some iterate, andhastwopositiveLyapunovexponents([V1],[AV]). Thesameisalsotruefor any f in a sufficiently small C3 neighbourhood of fˆ. N Letus fix somesmall α, andlet be asufficiently smallC3 neighbourhood of fˆsuch that for every f the aNbove properties hold. Choose some f ; ∈N ∈N if f is not mixing, we consider instead the first mixing power. Theorem 4 For ϕ ∞(S1 R,ν), ψ (R4,γ), we have estimates for (u ) n ∈ L × ∈ as in the indifferent fixed point case (Theorem 2) for every τ >1. The Central Limit Theorem holds for ψ (R4,γ) when γ is sufficiently ∈ large, depending on (ψ). ∞ R Anotherwayofsayingtheaboveisthatifψ (R4,γ),thenu = (n2−ζγ), n ∈ O withtheusualdependencyofζon (ψ). Notethatforobservablesin (R4,γ), ∞ γ>1 R ∩ we get super-polynomial decay of correlations, the same estimate as we obtain for Ho¨lder observables (though Baladi and Gou¨ezel have recently announced a stretched exponential bound for Ho¨lder observables - see BG). There are a number of generalisations we could consider, such as allowing D 2 ([BST] - note they require f to be C∞ close to fˆ), or replacing sin(2πs) ≥ by an arbitrary Morse function. 2.5 Non-uniformly expanding maps Finally,wediscussprobablythemostgeneralcontextinwhichourmethodscan currentlybe applied,thesettingof[ALP]. Inparticular,this settinggeneralises that of Viana maps. Let f :M M be a transitive C2 local diffeomorphismawayfrom a singu- → lar/critical set , with M a compact finite-dimensional Riemannian manifold. S Let Leb be a normalised Riemannian volume form on M, which we shall refer to as Lebesgue measure, and d a Riemannian metric. We assume f is non- uniformly expanding, or more precisely, there exists λ>0 such that n−1 1 liminf log Df−1 −1 λ>0. (1) n→∞ n k fi(x)k ≥ i=0 X For almost every x in M, we may define n−1 1 (x)=min N : log Df−1 −1 λ/2, n N . E ( n k fi(x)k ≥ ∀ ≥ ) i=0 X The decay rate of the sequence Leb (x)>n may be considered to give a {E } degree of hyperbolicity. Where is non-empty, we need the following further S assumptions, firstly on the critical set. We assume is non-degenerate, that is, C m( )= 0, and β > 0 such that x M we have d(x, )β . Df v / v . x C ∃ ∀ ∈ C C k k k k 6 d(x, )−β v T M, and the functions logdetDf and log Df−1 are locally x C ∀ ∈ k k Lipschitz with Lipschitz constant .d(x, )−β. C Now let d (x, ) = d(x, ) when this is δ, and 1 otherwise. We assume δ S S ≤ that for any ε>0 there exists δ >0 such that for Lebesgue a.e. x M, ∈ n−1 1 limsup logd (fj(x), ) ε. (2) δ n→∞ n − S ≤ j=0 X We define a recurrence time n−1 1 (x)=min N 1: logd (fj(x), ) 2ε, n N . δ T ( ≥ n − S ≤ ∀ ≥ ) i=0 X Letf beamapsatisfyingtheaboveconditions,andforwhichthereexistsα>1 such that Leb( (x)>n or (x)>n )= (n−α). {E T } O Thenf admitsanacipν withrespecttoLebesguemeasure,andwemayassume ν to be mixing by taking a suitable power of f. Theorem 5 For ϕ ∞(M,ν),ψ (R4,γ), we have estimates for (u ) as n ∈ L ∈ in the indifferent fixed point case (Theorem 2), for τ = α. Furthermore, when α > 2, the Central Limit Theorem holds for ψ (R4,γ) when γ is sufficiently ∈ large for given (ψ). ∞ R 3 Young’s tower In the previous section, we indicated the variety of systems we may consider. Weshallnowstatethemaintechnicalresult,andwithittheconditionsasystem must satisfy in order for our result to be applicable. As verifying that a system satisfies such conditions is often considerable work,we refer the reader to those papers mentioned in each of the previous subsections for full details. Therelevantsettingforourargumentswillbethetower objectintroducedby Youngin[Yo],andwerecapitsdefinition. WestartwithamapFR :(∆ ,m )(cid:9), 0 0 where (∆ ,m ) is a finite measure space. This shall represent the base of the 0 0 tower. We assume there exists a partition (mod 0) = ∆ : i N of ∆ , 0,i 0 P { ∈ } such that FR ∆ is an injection onto ∆ for each ∆ . We require that the 0,i 0 0,i partition gene|rates, i.e. that ∞ (FR)−j is the trivial partition into points. We also choose a return time fujn=c0tion R:P∆ N, which must be constant on 0 W → each ∆ . 0,i We define a tower to be any map F :(∆,m)(cid:9) determined by some FR, , P and R as follows. Let ∆ = (z,l) : z ∆ , l < R(z) . For convenience let ∆ 0 l { ∈ } refer to the set of points (,l) in ∆. This shall be thought of as the lth level · of ∆. (We shall freely confuse the zeroth level (z,0) : z ∆ ∆ with ∆ 0 0 { ∈ } ⊂ itself. We shall also happily refer to points in ∆ by a single letter x, say.) We 7 write ∆ = (z,l):z ∆ for l <R(∆ ). The partition of ∆ into the sets l,i 0,i 0,i { ∈ } ∆ shall be denoted by η. l,i The map F is then defined as follows: (z,l+1) if l+1<R(z) F(z,l)= (FR(z),0) otherwise. (cid:26) We noticethatthemapFR(x)(x)on∆ isidenticaltoFR(x), justifying our 0 choiceof notation. Finally, we define a notionofseparation time; for x,y ∆ , 0 ∈ s(x,y) is defined to be the least integer n 0 s.t. (FR)nx,(FR)ny are in ≥ different elements of . For x,y some ∆ , where x = (x ,l), y = (y ,l), we l,i 0 0 P ∈ set s(x,y):=s(x ,y ); for x,y in different elements of η, s(x,y)=0. 0 0 We say that the Jacobian JFR of FR with respect to m is the real-valued 0 function such that for any measurable set E on which JFR is injective, m (FR(E))= JFRdm . 0 0 ZE We assume JFR is uniquely defined, positive, and finite m -a.e. We require 0 some further assumptions. Measure structure: Let be the σ-algebra of m -measurable sets. We 0 • assume that all elementsBof and each n−1(FR)−i belong to , and P i=0 P B that FR and (FR ∆ )−1 are measurable functions. We then extend m 0,i 0 | W to a measure m on ∆ as follows: for E ∆ , any l 0, we let m(E) = l ⊂ ≥ m (F−lE), provided that F−lE . Throughout, we shall assume that 0 ∈ B anysetswechoosearemeasurable. Also,wheneverwesaywearechoosing an arbitrary point x, we shall assume it is a good point, i.e. that each elementofitsorbitis containedwithinasingleelementofthe partitionη, and that JFR is well-defined and positive at each of these points. Bounded distortion: There exist C > 0 and β < 1 s.t. for x,y any • ∈ ∆ , 0,i ∈P JFR(x) 1 Cβs(FRx,FRy). JFR(y) − ≤ (cid:12) (cid:12) (cid:12) (cid:12) Aperiodicity: Weassu(cid:12)methatgcd (cid:12)R(x):x ∆ =1. Thisisanecessary (cid:12) (cid:12) 0 • { ∈ } and sufficient condition for mixing (in fact, for exactness). Finiteness: We assume Rdm < . This tells us that m(∆)< . 0 • ∞ ∞ Let F :(∆,m)(cid:9) be a towRer, as defined above. We define classes of observ- able similar to those we consider on the manifold, but characterised instead in terms of the separationtime s on ∆. Given a bounded function ψ :∆ R, we → define the variation for n 0: ≥ v (ψ)=sup ψ(x) ψ(y) :s(x,y) n . n {| − | ≥ } Let us use this to define some regularity classes: 8 Exponential case: ψ (V1,γ), γ (0,1), if v (ψ)= (γn); n ∈ ∈ O Stretchedexponentialcase: ψ (V2,γ),γ (0,1),ifv (ψ)= (exp nγ ); n ∈ ∈ O {− } Intermediate case: ψ (V3,γ), γ >1, if v (ψ)= (exp (logn)γ ); n ∈ O {− } Polynomial case: ψ (V4,γ), γ >1, if v (ψ)= (n−γ). n ∈ O We shall see that the classes (V1-4) of regularity correspond naturally with the classes (R1-4) of regularity on the manifold respectively, under fairly weak assumptions on the relation between the system and the tower we construct for it. (We shall discuss this further in 13.) These classes are essentially those § definedin[P],althoughtherethefunctionsareconsideredtobepotentials rather than observables. We now state the main technical result. Theorem 6 Let F : (∆,m) (cid:9) be a tower satisfying the assumptions stated above. Then F : (∆,m) (cid:9) admits a unique acip ν, which is mixing. Further- more, for all ϕ,ψ ∞(∆,m), ∈L (ϕ Fn)ψdν ϕdν ψdν ϕ C(ψ)u , ∞ n ◦ − ≤k k (cid:12)Z Z Z (cid:12) (cid:12) (cid:12) where C(ψ)>(cid:12)0 is some constant, and (u ) is(cid:12)a sequence converging to zero at (cid:12) n (cid:12) some rate determined by F and v (ψ). In particular: n Case 1: Suppose m R>n = (θn), some θ (0,1). Then 0 { } O ∈ if ψ (V1,γ) for some γ (0,1), then u = (θn) for some θ (0,1); n • ∈ ∈ O ∈ ′ if ψ (V2,γ) for some γ (0,1), then u = (e−nγ ) for every γ′ <γ; n • ∈ ∈ O ′ if ψ (V3,γ) for some γ >1, then u = (e−(logn)γ ) for every γ′ <γ; n • ∈ O for any constant C > 0, there exists ζ < 1 such that if ψ (V4,γ) for ∞ • ∈ some γ > 1, and v (ψ)<C , then u = (n1−ζγ). ζ 0 ∞ n O Case 2: Suppose m R > n = (n−α) for some α > 1. Then for every 0 { } O C > 0 there exists ζ < 1 such that if ψ (V4,γ) for some γ > 2, with ∞ ∈ ζ v (ψ)<C , then 0 ∞ if γ = α+1, u = (n1−αlogn); • ζ n O otherwise, u = max(n1−α,n2−ζγ) . n • O The existence of a mi(cid:0)xing acip is proved(cid:1)in [Yo], as is the result in the case ψ (V1). As a corollary of the above, we get a Central Limit Theorem in the ∈ cases where the rate of mixing is summable. Corollary 1 Suppose F satisfies the above assumptions, and m R > n = 0 { } (n−α), for some α > 2. Then the Central Limit Theorem is satisfied for O ψ (R4,γ) when γ is sufficiently large, depending on F and v (ψ). 0 ∈ In 13 we shall give the exact conditions needed on a system in order to § apply the above results. 9 4 Overview of method Our strategy in proving the above theorem is to generalise a coupling method introducedby Young in[Yo]. Ourargumentfollowscloselythe line ofapproach of that paper, and we give an outline of the key ideas here. First, we need to reduce the problem to one in a slightly different context. Given a system F : (∆,m) (cid:9), we define a transfer operator F which, for any ∗ measure λ on ∆ for which F is measurable, gives a measure F λ on ∆ defined ∗ by (F λ)(A)=λ(F−1A) ∗ whenever A is a λ-measurable set. Clearly any F-invariant measure is a fixed point for this operator. Also, a key property of F is that for any function ∗ φ:∆ R, → φ Fdλ= φd(F λ). ∗ ◦ Z Z Next, we define a variation norm on m-absolutely continuous signed mea- sures, that is, on the difference between any two (positive) measures which are absolutely continuous. Given two such measures λ,λ′, we write dλ dλ′ λ λ′ := dm. | − | dm − dm Z (cid:12) (cid:12) (cid:12) (cid:12) Now let us fix an acip ν and choose o(cid:12)(cid:12)bservables (cid:12)(cid:12)ϕ ∞(∆,ν), ψ 1(∆,ν), ∈ L ∈ L with infψ >0, ψdν =1. We have R (ϕ Fn)ψdν = (ϕ Fn)d(ψν) ◦ ◦ Z Z = ϕd(Fn(ψν)), ∗ Z where ψν denotes the unique measure which has density ψ with respect to ν. So dFn(ψν) dν (ϕ Fn)ψdν ϕdν ψdν ϕ ∗ dm ∞ ◦ − ≤ k k dm − dm (cid:12)Z Z Z (cid:12) Z (cid:12) (cid:12) (cid:12) (cid:12) = ϕ Fn(cid:12)(ψν) ν . (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) k k∞| ∗(cid:12)(cid:12) − | (cid:12)(cid:12) Hence we may reduce the problem to one of estimating the rate at which certain measures converge to the invariant measure, in terms of the variation norm. Infact,itwillbeusefultoconsiderthemoregeneralquestionofestimat- ing Fnλ Fnλ′ for a pairof measuresλ, λ′ whose densities with respect to m | ∗ − ∗ | are of some given regularity. (We shall require an estimate in the case λ′ = ν 6 when we consider the Central Limit Theorem.) Let us now outline the main argument. We work with two copies of the system, and the direct product F F : (∆ ∆,m m) (cid:9). Let P = λ λ′, 0 × × × × 10