Functional Norms for Young Towers

Transcription

1 Fairfield University Mathematics Faculty Publications Mathematics Deartment Functional Norms for Young Towers Mark Demers Fairfield University, Coyright 2010 by Cambridge University Press. Original ublished version can be found at DOI: /S Peer Reviewed Reository Citation Demers, Mark, "Functional Norms for Young Towers" (2010). Mathematics Faculty Publications. 9. htt://digitalcommons.fairfield.edu/mathandcomuterscience-facultyubs/9 Published Citation M. Demers, Functional Norms for Young Towers, rgodic Theory and Dynamical Systems. 30 (Part 5), This Article is brought to you for free and oen access by the Mathematics Deartment at It has been acceted for inclusion in Mathematics Faculty Publications by an authorized administrator of For more information, lease contact

2 rgod. Th. & Dynam. Sys. (2010), 30, doi: /s c 2009 Cambridge University Press Printed in the United Kingdom Functional norms for Young towers MARK F. DMRS Deartment of Mathematics, Fairfield University, Fairfield CT 06824, USA ( mdemers@fairfield.edu) (Received 15 January 2009 and acceted in revised form 22 May 2009) Abstract. We introduce functional norms for hyerbolic Young towers which allow us to directly study the transfer oerator on the full tower. By eliminating the need for secondary exanding towers commonly emloyed in this context, this aroach simlifies and exands the analysis of this class of Markov extensions and the underlying systems for which they are constructed. As an examle, we rove large-deviation estimates with a uniform rate function for a large class of non-invariant measures and show how to translate these to the underlying system. 1. Introduction Young towers were introduced in [Y1] as a unified framework in which to view the statistical roerties of both uniformly and non-uniformly hyerbolic dynamical systems. Briefly, given a dynamical system f : M, a Young tower is a tye of Markov extension F : with the following reresentation. Given a reference measure µ, one chooses a reference set M of ositive measure with a hyerbolic roduct structure and constructs a return time function R : Z + with certain (Markov) roerties. The Young tower is an extension of l 0 f l where the lth level of the tower corresonds to those x f l for which R(x) > l. In essence, the Young tower reresents as a horseshoe with countably many branches and variable return times. The rate of decay in the measure of the levels of the tower, µ(r(x) > l), gives information about the statistical roerties of f ; for examle, it reflects the exonential or olynomial decay of correlations (see [CY] for a survey). Young towers have been constructed for many systems: billiards with convex scatterers, including those subject to external forces [C3, C2, Y1]; iecewise hyerbolic attractors [C1, Y1]; Hénon mas [BY, Y1]; non-uniformly exanding mas in one dimension [WY, Y1]; and Lorentz attractors [HM]. The strategy in all these aers is the same. One first constructs a Young tower F : satisfying certain roerties (see (P1) (P5) of 2.1). One then defines a quotient tower = / where x y whenever x and y lie on the same stable leaf. There are thus three

3 1372 M. F. Demers objects involved in the study of such systems. Letting π and π reresent the canonical rojections from (F, ) to ( f, M) and (F, ) resectively, we reresent them as follows: F : π π f : M F : The reason for introducing the reduced tower is that since F : is exanding, one can bring to bear the classical methods of analysis of the transfer oerator for exanding mas. This functional analytic aroach establishes the quasi-comactness of the transfer oerator acting on certain saces of functions and then links its eriheral sectrum to the statistical roerties of the system [B, DF, HH, IM, K, LY, N]. In the case of Young towers, one must then ass these statistical roerties from back u to before rojecting them down to M. The roblem is that in general one cannot lift measures from to so that the usual rocedure is to first rove the existence of an invariant measure on searately (see [Y1]) and then ass from to the desired roerties related to the invariant measure such as central limit theorems, decay of correlations or large-deviation estimates (cf. [MN, RY]). From, these roject easily onto M. The urose of this aer is to simlify the alication of Young towers by directly studying the transfer oerator L associated with (F, ), thus eliminating the need for the reduced tower (F, ) entirely. We introduce Banach saces on which L is quasicomact and obtain its sectral decomosition, following the recent extensions of this method to the hyerbolic setting [BT, BKL, DL, GL, R]. In doing so, we recover the statistical roerties which have been roven for f : M reviously. In addition, we are able to obtain much more information about the evolution of non-invariant measures under the dynamics of f. We include as an examle a large-deviation rincile for a large class of initial distributions which is entirely new in the non-uniformly hyerbolic setting (Theorems 4 and 5) and include an exlicit alication to disersing billiards (Theorem 6). We show that the rate function governing the large deviations is indeendent of the initial distribution. It may be of some indeendent interest that when no contracting directions are resent, i.e., when f itself is exanding, the norms we define for hyerbolic towers reduce to norms for exanding towers which yield analogous results: i.e., the quasi-comactness of the transfer oerator and a sectral ga. This is a characteristic of our norms which has not been resent reviously in the hyerbolic setting and which yields a unified treatment of hyerbolic and exanding Young towers (see Remark 2.2 for more details). The rest of this aer is organized as follows. In 2, we define Young towers recisely, define the relevant norms and state our results. In 3, we exlore some roerties of the Banach saces while 4 and 5 contain the required Lasota Yorke tye estimates and sectral results. Large-deviation estimates and alications are roved in 6 and Setting and statement of results 2.1. Definition of the tower. We recall the definition of a Young tower (F, ) as described in [Y1]. We begin with a iecewise smooth ma on a finite-dimensional

4 Functional norms for Young towers 1373 Riemannian manifold, f : M, and let µ (resectively µ γ ) denote Riemannian volume on M (resectively γ where γ M is a submanifold). We say that f admits a generalized horseshoe if there exists a comact subset of M satisfying roerties (P1) (P5) below. (We recall only the main roerties; see [Y1] for more details.) (P1) Hyerbolic roduct structure. = J j=1 ( j) and each of the ( j) has the following roduct structure : ( j) = ( γ u Ɣ u γ u ) ( j γ s Ɣ s γ s ) where Ɣ u = j j Ɣu j and Ɣ s = j Ɣs j are continuous families of local stable and unstable manifolds such that, for every j, each γ u Ɣ u j intersects every γ s Ɣ s j in a unique oint. Moreover, µ γ (γ ) > 0 for each γ Ɣ u. A set A is an s-subset (resectively u-subset) of if γ A imlies γ A for any γ Ɣ s(u). (P2) Return time function. ach ( j) is artitioned into countably many s-subsets ( j) i with µ γ ( ( j) \ j) i ( i ) = 0 for all γ Ɣ u j. There exists a function, R : Z+, constant on each ( j) i, such that f i ) is a u-subset of one of the (k). Moreover, for each n, the number of (i, j) such that R ( j) i = n is finite. We refer to elements of Ɣ u(s) by γ u(s) and J u f denotes the unstable Jacobian of f with resect to µ γ u. Denote by γ s (x) and γ u (x) the stable and unstable leaves through x, resectively. For x, y, there exists a searation time s 0 (x, y), deending only on the unstable coordinate, and numbers C 0 1, α < 1 indeendent of x, y, such that the following hold. (P3) Contraction on Ɣ s. For y γ s (x), d( f n x, f n y) C 0 α n d(x, y) for all n 0. (P4) Backward contraction and distortion on Ɣ u. Let y γ u (x) and 0 k n < s 0 (x, y). (a) d( f n x, f n y) C 0 α s0(x,y) n. (b) log n i=k J u f ( f i x)/j u f ( f i y) C 0 α s0(x,y) n. (P5) Convergence of J u f n and absolute continuity of Ɣ s. (a) For y γ s (x) and all n 0, log R( ( j) i ) ( ( j) i=n J u f ( f i x) J u f ( f i y) C 0α n. (b) Given γ, γ Ɣ u, define : γ γ by (x) = γ s (x) γ. Then is absolutely continuous and d( 1 µ γ ) = dµ γ i=0 J u f ( f i x) J u f ( f i x). The structure of the generalized horseshoe immediately yields the existence the Young tower, = {(x, n) N : n < R(x)}. In [Y1], the tower has a single base and all returns are full returns to the base. In the resent aer we treat the generalized case of towers with multile bases. These are the ( j). As an abstract requirement, (P3) is slightly stronger than the inequality d( f n x, f n y) C 0 α n stated in [Y1]. In ractice, however, the stronger version holds for all systems for which Young towers have been constructed to date.

5 1374 M. F. Demers The tower ma F is defined by F(x, l) = (x, l + 1) for l < R(x) 1 and F(x, R(x) 1) = ( f R(x) x, 0). We refer to l := n=l as the lth level of the tower and to a oint (x, l) as simly x where the level l will be made clear by context. F admits a Markov artition { l, j } which has finitely many elements on each level l. We identify 0, j with ( j) for j = 1,..., J. We define a searation time s(x, y) s 0 (x, y) by s(x, y) = inf{n > 0 : F n x, F n y lie in different l, j }. There is a canonical rojection π : M which conjugates the dynamics, π F = f π. π is not necessarily one-to-one or onto, but if f is injective, then so is π l for each l. We call F transitive if, for all j, j, there exists n 0 such that F n ( 0, j ) 0, j. We say F is mixing if, for all j, j, there exists N 0 such that F n ( 0, j ) 0, j for all n N. Similarly, we call T a transitive comonent if T is a union of elements l, j such that F T is transitive. The base of the tower, 0, is identified with and inherits the structure of Ɣ s and Ɣ u as well as the measures µ and µ γ. The measure µ is extended to l, l > 0, by defining µ(a) = µ(f l A) for all measurable A l. Thus J F, the Jacobian of F with resect to µ, satisfies J F 1 excet at return times. F enjoys roerties (P3) (P5) at return times due to the identity Jπ(F)J F = J f (π)jπ and the fact that Jπ 1 on 0. Since F : l l+1 is rigid translation, the foliations Ɣ s and Ɣ u extend naturally to the entire tower, i.e., l, j has a roduct structure given by F l (S) where S is an s- subrectangle in 0, and hence in. We call these defining foliations Ɣ s ( ) and Ɣ u ( ). We say the tower has exonential return times if the following holds: ( ) there exist constants c 0 > 0 and θ < 1 such that µ(r > n) = µ l c 0 θ n. (1) xonential return times are essential to our aroach since we will rove the existence of a sectral ga which in turn imlies exonential decay of correlations. Towers with olynomial return times have been shown to admit olynomial decay of correlations [Y2]. Accordingly, we will assume that (1) holds throughout this aer. l n A reference measure on. In each l, j we choose a reresentative curve γ Ɣ u ( l, j ). For any γ Ɣ u ( l, j ), let γ, γ : γ γ denote the holonomy ma along Ɣ s -leaves and let J u F denote the unstable Jacobian of F with resect to µ γ. Then define m γ by dm γ = φ dµ γ where φ(x) = i=0 J u F(F i x) J u F(F i ( γ, γ x)). Given γ Ɣ u ( 0 ), if γ Ɣ u ( l, j ) satisfies F(γ S) = γ for some s-subrectangle S, then for x γ S, define J γ F(x) = d(m γ F)/dm γ. lsewhere on, J γ F 1. Similarly, one defines J γ F R (x) whenever F R (γ S) = γ for some s-subrectangle S. For convenience, we restate Lemma 1 from [Y1], which summarizes the imortant roerties of m γ. For mas with singularities, the elements of Ɣ s ( ) and Ɣ u ( ) are actually ositive-measure Cantor sets in real stable and unstable manifolds for f, as described in [Y1]. For simlicity, we shall refer to these Cantor sets as stable and unstable leaves throughout.

6 Functional norms for Young towers 1375 LMMA 2.1. [Y1] Let γ, γ Ɣ u ( l, j ). (1) Let γ,γ : γ γ be the holonomy ma along Ɣ s -leaves as above. Then m γ = m γ. (2) J γ F(x) = J γ F(y), for all x γ, y γ s (x) γ. (3) There exists C 1 > 0 such that for all x, y γ with s 0 (x, y) R(x), J γ F R (x) J γ F R (y) 1 C 1α s(f R x,f R y)/2. Moreover by (P5)(a), e C 0 φ e C 0. On each l, j, we define the measure µ s on Ɣ u ( l, j ) to be the factor measure of µ l, j on unstable leaves normalized so that µ s (Ɣ u ( l, j )) = 1. We define m to be the measure with factor measure µ s and measures m γ on unstable leaves. Notice that in any l, j, Lemma 2.1(1) imlies that m γ (S) = m(s) for any s-subset S l, j and γ Ɣ u ( l, j ). This feature of m γ imlies that m is a roduct measure on each l, j. When disintegrating m, we maintain the convention that µ s is normalized, but m γ is not Transfer oerator. The rimary object of interest in this aer is the transfer oerator L associated with F. Before defining it, we introduce a class of functions. We define a metric along stable leaves which makes the distance between unstable leaves uniform. Fix x 0 and let y γ s (x). Let : γ u (x) γ u (y) be the sliding ma along stable leaves. Define d s (x, y) := su z γ u (x) d(z, z). We extend this metric to l, l > 1, by setting d s (F l x, F l y) = α l d s (x, y) for all l < R(x) and y γ s (x). By (P3), d s (F n x, F n y) C 0 α n d s (x, y) for all n 0 whenever y γ s (x). (2) The class of test functions we use are required to be smooth along stable leaves only. Let F b denote the set of bounded measurable functions on. For ϕ F b and 0 < r 1, define H r s (ϕ) = su γ s Ɣ s ( ) H r (ϕ γ s ) where H r (ϕ γ s ) = su x,y γ s ϕ(x) ϕ(y) d s (x, y) r. If A is an s-subset of, we define ϕ C r s (A) = su γ s A ϕ C 0 (γ s ) + Hr (ϕ γ s ) and let C r s (A) = {ϕ F b : ϕ C r s (A) < }. For h (C r s ( )) an element of the dual of C r s ( ), the transfer oerator L : (Cr s ( )) (C r s ( )) is defined by Lh(ϕ) = h(ϕ F) for each ϕ C r s ( ). When h is a measure absolutely continuous with resect to the reference measure m, we shall call its L 1 (m) density h as well. Hence h(ϕ) = hϕ dm. With this convention, L 1 (m) (Cs r ( )) and one can restrict L to L 1 (m). In this case, L n h(x) = h(y)(j m F n (y)) 1 y F n x for each n 0 where J m F n is the Jacobian of F n with resect to m. Along unstable leaves, we define the metric d u (x, y) = β s(x,y) 0 for y γ u (x) and some β 0 < 1 to be chosen later. Let Li u (ϕ γ ) denote the Lischitz constant of a function ϕ along γ Ɣ u with resect to d u and define Li u (ϕ) = su γ Ɣ u ( ) Liu (ϕ γ ). We define Li u ( ) = {ϕ F b : Li u (ϕ) < }.

7 1376 M. F. Demers 2.2. Definition of norms. Let P = { l, j } denote the Markov artition for F. For each k 0, define P k = k i=0 F i P and let P k l, j = Pk l, j. The elements P k l, j are k-cylinders which are s-subsets of l, j. For ψ L 1 (m) and P k, define ψ dm = 1 ψ dm. m() Now choose 0 < q < 1 and fix 1 > β 0 > max{θ, α} where θ is given by (1) and α is from (P3). Next, choose 1 > β max{β ( q)/ 0, α q }. For h Li u ( ), define the weak norm of h by h w = su l, j,k h w(p k l, j ) where h w(p k l, j ) = βl 0 su P k l, j su hϕ dm. (3) ϕ Cs () 1 Define the strong stable norm of h by h s = su l, j,k h s(p k l, j ) where h s(p k l, j ) = βl su P k l, j su hϕ dm. (4) ϕ q Cs () 1 For ϕ C s ( ), define ϕ on P k l, j by ϕ (x) = m() 1 γ u (x) ϕ dm γ, for x. Let ϕ (x) = ϕ (γ u (x)) for x l, j be the extension of ϕ to l, j. Note that ϕ is welldefined since ϕ is constant on unstable leaves. In what follows, let k P k l, j, r P r l, j for r k. We define the strong unstable norm of h by h u = su l, j,k h u(p k l, j ) where h u(p k l, j ) = su su su β l k hϕ dm k Pl, k r k ϕ j Cs (r ) 1 r h ϕ r dm. (5) k The strong norm of h is defined as h = h s + b h u, for some b > 0 to be chosen later. We denote by B the comletion of Li u ( ) in the -norm and by B w the comletion of Li u ( ) in the w norm. Remark 2.2. If there is no stable direction, on each l, j the weak norm, w, reduces to the C 0 norm of h weighted by β0 l. Similarly, the strong stable and unstable norms reduce to the C 0 norm and Lischitz constant of h resectively, each weighted by β l. The Lasota Yorke estimates (Proosition 2.3) and the comactness argument (Lemma 3.6) both hold in this setting so that one immediately obtains the results of Theorem Statement of results on the tower. The standing assumtions throughout this aer are that (F, ) satisfies (P1) (P5) and (1). PROPOSITION 2.3. There exists C > 0 such that, for each h B and n 0, L n h w C h w, (6) L n h s Cβ n h s + C h w, (7) L n h u Cβ n h u + C h s. (8)

8 Functional norms for Young towers 1377 For any 1 > τ > β, there exists N 0 such that 2Cβ N < τ N. Choose b = β N. Then, L N h = L N h s + b L N h u Cβ N ( h s + b h u ) + bc h s + C h w τ N h + C h w. The above reresents the traditional Lasota Yorke inequality. Since by Lemma 3.6, the unit ball of B is relatively comact in B w, it follows from standard arguments [DF, B] that the essential sectral radius of L on B is bounded by β. Our first theorem resents the decomosition of the eriheral sectrum. THORM 1. The oerator L : B is quasi-comact with essential sectral radius β and sectral radius 1. In addition, the following hold. (i) If F is mixing, then 1 is a simle eigenvalue and all other eigenvalues have modulus strictly less than 1. (ii) If F is transitive and eriodic with eriod, then the set of eigenvalues of modulus 1 consists of simle eigenvalues {e 2πik/ } 1 k=0. (iii) In general, F has finitely many transitive comonents, each with largest eigenvalue 1. On each comonent, (ii) alies. Let V φ be the eigensace of L corresonding to the eigenvalue e iφ and set V := φ V φ. Our next results characterize the set of invariant measures in B and some of the statistical roerties of F. Let Cb 0 ( ) denote the set of bounded functions on which are continuous on each l, j. Recall that an invariant robability measure ν is called a hysical measure if there exists a ositive Lebesgue measure invariant set B ν, with ν(b ν ) = 1, such that, for each ψ Cb 0, 1 n 1 lim ψ(f i x) = ν(ψ) for all x B ν. n n i=0 THORM 2. (i) ach ν V is a signed measure absolutely continuous with resect to the robability measure ν := lim n (1/n) n 1 i=0 Li 1. The conditional measures of ν on γ Ɣ u are absolutely continuous with resect to µ γ. (ii) F admits only finitely many hysical measures and they are recisely the ergodic elements of V 0. The suorts of the hysical measures corresond to the ergodic decomosition with resect to Lebesgue and ν() = m() for each ergodic comonent. (iii) For all ψ Cb 0( ) and every γ Ɣu ( ), the limit ψ + (x) := lim n (1/n) n 1 i=0 ψ F i (x) exists for µ γ -almost every x γ and takes on only finitely many different values in. If ν is ergodic, then ψ + (x) = ψ dν for µ γ -almost every x. (iv) If F is mixing, then F exhibits exonential decay of correlations for Hölder observables, and the central limit theorem holds. In articular, there exist constants C 1 > 0, σ < 1, such that for any ψ Li u ( ) and ϕ Cs ( ), ψϕ F n dν ν(ψ)ν(ϕ) C 1σ n ϕ Cs ( ) ( ψ + Li u (ψ)).

9 1378 M. F. Demers Large-deviation estimates. An immediate alication of the sectral decomosition of L is the derivation of large-deviation estimates for smooth observables g : R. Let f : X be a measurable ma. Letting S n g = n 1 k=0 g f k, if ν is an ergodic invariant measure, then (1/n)S n (g) converges to ν(g) by the Birkhoff ergodic theorem. Large-deviation estimates rovide exonential bounds on the rate of convergence of S n g to the mean ν(g). These tyically take the form ( lim lim ν x : 1 ) ε 0 n n S n(g)(x) [t ε, t + ε] = I (t) (9) where I (t) 0 is called the rate function. Large-deviation estimates of this tye have been roved for systems admitting Young towers [RY, MN]. When f is (non-uniformly) exanding and ν is has strictly ositive density with resect to Lebesgue, the measure on the left-hand side of (9) can be relaced by Lebesgue [KN]. This is also true when f is Axiom A [OP]. For more general nonuniformly hyerbolic systems, (9) is not known to be true when ν is not an invariant measure. In the setting of the resent aer, we rove as a direct corollary of our oerator aroach that (9) holds for all robability measures in B with the same rate function I. Let σ 2 denote the limit as n of the variance of (1/ n)s n g with resect to the unique invariant measure for F. THORM 3. Let (F, ) be mixing and let g Li u ( ) Cs ( ). There exist constants τ max, ω max > 0 such that for all robability measures η B, the logarithmic moment generating function 1 q(z) = lim n n log η(ezsng ) exists, is indeendent of η and is analytic in the rectangle {z C : Re z < τ max, Im z < ω max }. Moreover, q (0) = ν(g), q (0) = σ 2 and q(z) is strictly convex for real z whenever σ 2 > 0. An immediate consequence of this theorem is a large-deviation result for robability measures in B. THORM 4. Let η B be a robability measure and let I (u) be the Legendre transform of q(z). Then, for any interval [a, b] [q ( τ max ), q (τ max )], ( 1 lim n n log η x : 1 ) n S ng(x) [a, b] = inf I (u). u [a,b] 2.4. Discussion of alications. Throughout this section, f : M is a ma which admits a Young tower F : as described in 2.1 with exonential return times. Let Ɣ s(u) (M) denote the set of local stable (unstable) manifolds on M. For ζ 0, C ζ (Ɣ u (M)) denotes the set of functions ϕ on M which satisfy su γ Ɣ u (M) ϕ C ζ (γ ) <, and similarly for C ζ (Ɣ s (M)).

10 Functional norms for Young towers 1379 THORM 5. (i) The ma f has at least one and at most finitely many hysical measures which lift to. ach of these hysical measures has absolutely continuous conditional measures on unstable leaves. For (ii) (iv), we assume f is mixing and that ξ π B is a robability measure. (ii) The ma f has a unique hysical measure ν = π ν. ν satisfies the central limit theorem and enjoys exonential decay of correlations, i.e., there exists C 2 > 0 such that ψϕ f n d ν ν(ψ) ν(φ) C 2σ n ψ C ζ (Ɣ u (M)) ϕ C ζ (Ɣ s (M)) for all ψ C ζ (Ɣ u (M)), ϕ C ζ (Ɣ s (M)), with σ as in Theorem 2(iv). (iii) (iv) The limit lim n f n ξ = ν weakly. Suose g C ζ (M) for some ζ > 0 and let S n g = n 1 i=0 g f i. Then the logarithmic moment generating function q(z) = lim n (1/n) log ξ(e S n g ) exists and satisfies the conclusions of Theorem 3. As a consequence, all robability measures ξ π B satisfy the large-deviation rincile given by Theorem 4 with the same rate function I. The discussion of which measures on M lift to, i.e., are in π B, deends on the roerties of the underlying system ( f, M). For non-uniformly exanding systems, one can show that Lebesgue measure µ π B and indeed C ζ (M) π B under fairly mild assumtions (cf. [BDM]). For hyerbolic systems, the question is more subtle. One can easily see that measures suorted on π( l, j ) for finitely many l, j with smooth conditional densities on Ɣ u (M) lift to measures in B. One may also allow singular measures suorted on a single local unstable leaf. For more general measures the story is not so simle and in general one cannot even guarantee that Lebesgue measure lifts to. One class of measures which always lift to are those measures which satisfy ξ = ψ ν for some ψ C ζ (Ɣ u (M)), for then defining η := (ψ π)ν, one has η B and π η = ξ (see, for examle, [DWY, Lemma 6.3]). For systems whose invariant measure is smooth with resect to Lebesgue, such as disersing billiards, we obtain convergence results for a class of measures which do not lift to, but can be aroximated by elements of π B. THORM 6. Let ( f, M) be the billiard ma corresonding to a disersing billiard which admits a mixing Young tower with exonential tail bounds as in [Y1, C3]. Let µ be Lebesgue measure and let ν be the smooth invariant measure for f. Let G denote the set of robability measures ξ on M such that dξ/dµ C ζ (Ɣ u (M)) for some ζ > 0. Let ξ G π B be a robability measure. Then the following hold. (i) Convergence to ν. f n ξ converges weakly to ν as n. (ii) Large deviations. For any g C ζ (M), ξ satisfies the large-deviation estimate of Theorem 4 with rate function I indeendent of ξ. 3. Proerties of the saces B and B w Before exloring the generalized function saces we have defined, we record the following lemma for future use.

11 1380 M. F. Demers LMMA 3.1. (Regularity of ϕ ) Let 0 < r 1 and k 0. For P k l, j and ϕ C r s (), let ϕ be defined as in 2.2. Then ϕ C r s () and ϕ C r s () ϕ C r s (). Proof. Choose Pl, k j and ϕ Cr s (). By definition, ϕ (x) = m() 1 γ u (x) ϕ dm γ. This immediately imlies ϕ ϕ. Take x, y γ s and let : γ u (x) γ u (y) denote the holonomy ma along stable leaves. Note that J 1 by Lemma 2.1(1). Then ϕ (x) ϕ (y) m() 1 (ϕ ϕ ) dm γ Hs r (ϕ)d s(x, y) r, γ u (x) using the fact that m γ () = m() and d s (z, z) = d s (x, y) for all z γ u (x) mbeddings. LMMA 3.2. There exists C > 0 such that for all h B w and ϕ C s ( ), we have h(ϕ) C h w ( ϕ + H s (ϕ)). Proof. Let h Li u ( ) and ϕ Cs ( ). Then hϕ dm = hϕ dm m( l, j )β l 0 h w ϕ Cs ( l, j ) l, j l, j l, j C h w ( ϕ + Hs (ϕ)), since β 0 > θ and m( l, j ) e C 0µ( l, j ) by Lemma 2.1. It is clear from the definition of u that measures h B necessarily have absolutely continuous conditional measures on h-almost-every γ Ɣ u ( ). Lemma 3.3 rovides some examles of what tyes of measures are found in B. Given a measure η with absolutely continuous conditional measures on unstable leaves, we define a measure η s on Ɣ u ( ), i.e., a measure transverse to unstable leaves, as follows: set η s (Ɣ u ( l, j )) = 0 if η l, j 0. If η l, j 0, then η s Ɣ u ( l, j ) is the factor measure of η l, j normalized, and {ρ γ dm γ, γ Ɣ u ( l, j )} is the disintegration of η into measures on unstable leaves. We will use the convention that η s ( l, j ) = 1, and that the densities ρ γ are not normalized. We define G to be the set of such measures η 0 whose (unnormalized) densities satisfy su γ Ɣ u ( ) ρ γ C 0 (γ ) + Liu (ρ γ ) <. LMMA 3.3. (i) G B and in articular Li u ( ) B. (ii) If h Li u ( ) Cs ( ) and g B, then hg B, hg s g s h C q s ( ) and hg u g s (Li u (h) + H s (h)) + g u h C s ( ). Proof. (i) We first show Li u ( ) B and then use these functions to aroximate singular measures η in the -norm. Let h Li u ( ). It follows immediately that h s h.

12 Functional norms for Young towers 1381 Now let k P k l, j and take r P r l, j with r k. For ϕ C s ( r ), define ϕ r as in 2.2. Choose γ s 0 Ɣs ( k ) and let h 0 (x) = h(γ u (x) γ s 0 ) for x l, j. Then r hϕ dm k h ϕ r dm = r (h h 0 )ϕ dm + r h 0 ϕ dm k h 0 ϕ r dm + k (h 0 h) ϕ r dm. (10) The first term of (10) is Li u (h)β k ϕ q Cs ( r ). The fourth term has the same bound since ϕ r q Cs ( k ) ϕ Cs q ( r ) by Lemma 3.1. Note that by Lemma 2.1(1), m γ (γ k ) = m( k ) for each γ Ɣ u ( l, j ) since µ s is a robability measure. Since h 0 and ϕ r are both constant on unstable leaves, h 0 ϕ r dm = m( k ) 1 dµ s (γ )h 0 ϕ r m γ ( k ) = k Ɣ u ( l, j ) h 0 ϕ dm, r so that the second and third terms of (10) cancel. Thus h u 2Li u (h) and h B. To rove the result for more general η G, it suffices to rove it for measures suorted on a single unstable leaf. More general measures follow by aroximation. Fix γ 0 Ɣ u ( l, j ) and let ρ be a Lischitz function on γ 0. Let η = ρm γ0. We take a sequence of smooth functions ψ n, deending only on the stable coordinate, which converge in distribution to δ γ0. xtend ρ to l, j by making it constant on stable leaves and define h n = ψ n ρ. Note that h n Li u ( ) Cs ( ) and h n clearly converges to η in distribution. It remains to show that {h n } is a Cauchy sequence in the -norm. For Pl, k j and ϕ Cq s (), (h n h m )ϕ dm = 1 dµ s (γ )(ψ n (γ ) ψ m (γ )) ρϕ dm γ. m() Ɣ u ( l, j ) γ Since m() 1 γ ρϕ dm γ is a Hölder continuous function of γ with C q norm bounded by ρ ϕ q Cs (), the integral above converges to 0 as n, m. Moreover, since the Cq norms are uniformly bounded for ϕ q Cs () 1, we may take the suremum and conclude convergence in the s -norm. To estimate h n h m u, let r k l, j and for ϕ Cs ( l, j ), define ϕ r as in 2.2. Choose γ s Ɣ s ( r ) and let ρ(x) = ρ(γ u (x) γ s ) for x l, j. Now set g n,m = h n h m and g n,m = (ψ n ψ m )ρ. Then following (10), since g n,m is constant along unstable leaves, we have g n,m ϕ dm g n,m ϕ r dm = (g n,m g n,m )ϕ dm + (g n,m g n,m ) ϕ r dm. r k r k (11) The first term above is equal to m( r ) 1 dµ s (γ )(ψ n ψ m ) γ r (ρ ρ)ϕ dm γ. Dividing by β k and using the fact that β k (ρ ρ) is bounded on r since ρ Li u ( l, j ), we see that the integral of interest has the form dµ s (γ )(ψ n ψ m ), where is a Hölder continuous function of γ with Hölder norm bounded by Li u (ρ) ϕ C s ( l, j ). We conclude that the integral converges to zero as n, m, uniformly for ϕ C s ( l, j ) 1. A similar estimate holds for the second term of (11) so that h n h m u 0 as n, m.

13 1382 M. F. Demers (ii) By density of Li u ( ) B, it suffices to rove the claim for h Li u ( ) Cs ( ) and g Li u ( ). That hg s g s h q Cs ( ) is immediate. Let r k, ϕ Cs ( ) and ϕ r be as above. We define h 0 as in (i) and follow (10) to obtain r hgϕ dm k gh ϕ r dm = r (h h 0 )gϕ dm + r h 0 gϕ dm k h 0 g ϕ r dm + k (h 0 h)g ϕ r dm. (12) To estimate the second and third terms of (12), we use the fact that h 0 C s ( l, j ) and (h 0 ϕ) r = h 0 ϕ r since h 0 is already constant on unstable leaves, where (h 0 ϕ) r denotes the average of h 0 ϕ on unstable leaves in r. Then r gh 0 ϕ dm k gh 0 ϕ r dm β k l g u h C s ( l, j ) ϕ C s ( l, j ). The first term of (12) is β l g s h h 0 q Cs ( k ) ϕ Cs q ( k ) and similarly for the fourth term. We must show h h 0 q Cs ( ) has order βk. Clearly, h h 0 Li u (h)β k 0 since the searation time in k is k. To estimate the Hölder norm, let x, y γ s. Then h(x) h 0 (x) h(y) h 0 (y) 2Li u (h)β k 0, estimating the x and y differences searately. On the other hand, since h C s ( ), h(x) h 0 (x) h(y) h 0 (y) 2H s (h)d s (x, y). The Hölder constant is bounded by the minimum of the two estimates 2 Li u (h)β k 0 d s(x, y) q and 2H s (ϕ)d s (x, y) q. This minimum is largest when the two estimates are equal, i.e., when β k 0 = d s(x, y). Thus H q s (h h 0 ) Cβ k( q)/ 0 Cβ k since β was chosen β ( q)/ 0. Remark 3.4. Since w s, there exists a natural embedding of B into B w. Moreover, Lemmas 3.2 and 3.3 imly that Li u ( ) B B w C s ( ). In fact, the inclusions are injective u to modification of h Li u ( ) on sets of m-measure zero. This can be roven as in [GL, Proosition 4.1]. Accordingly, we will consider B as a subset of B w and Li u ( ) as a subset of B by identifying h Li u ( ) with the measure hm Comactness. LMMA 3.5. On a fixed P k 0 l, j, the unit ball of is comactly embedded in w. Proof. Let ε > 0 be fixed. We will show that there are finitely many ψ i Cs () such that, for any h Li u ( ) and any ϕ Cs () with ϕ Cs () 1, there exists an i such that β0 l hϕ dm h ψ i dm b 1 h ε.

14 Functional norms for Young towers 1383 Now let ϕ Cs () be such that ϕ Cs () 1. For k k 0, let i k denote the finitely many k-cylinders in. For ϕ Cs r () and x k i, define ϕ k,i (x) = m(i k ) 1 ϕ dm γ, γ u (x) k i and note that ϕ k,i Cs r (k i ) by Lemma 3.1. Let Hr (k, i) be the sace of such averaged Hölder functions generated by ϕ Cs r (). Since > q, it is a standard consequence of the Arzela Ascoli theorem that the unit ball of H (k, i) is comactly embedded in Hq (k, i) for each i 0, k k 0. Thus we may choose finitely many functions {ψ k,i,n } n=1 N H (k, i) which form an ε-covering in the Cq s -norm of the unit ball of Cs (i k). Choose k such that β k < ε and let ϕ k = i ϕ k,i1 k. On each k i i, choose 1 n N such that ψ k,i,n ϕ k,i q Cs (i k) ε and let ψ k = i ψ k,i,n1 k. Then ϕ k, ψ k Cs () by i Lemma 3.1 and ψ k ϕ k q Cs () ε. Now, h ϕ dm hψ k dm h(ϕ ϕ k ) dm + h(ϕ k ψ k ) dm. (13) We estimate the first term of (13) using the strong unstable norm, h(ϕ ϕ k ) dm = m(i k ) h ϕ dm h ϕ k,i dm i k m()βk l h u. i i k We estimate the second term of (13) using the strong stable norm, h(ϕ k ψ k ) dm m()β l h s ϕ k ψ k q Cs () m()β l ε h s. Then since β 0 < β and β k < ε, we have β0 l hϕ dm hψ k dm βl 0 βk l h u + β0 l β l ε h s b 1 h ε. LMMA 3.6. The unit ball of B is comactly embedded in B w. Proof. In light of Lemma 3.5, it suffices to show that the weak norm of h B can be aroximated by considering its norm on only finitely many k-cylinders Pl, k j. This will follow from the fact that β 0 < β and the averaging roerty of the strong unstable norm. Notice that for h Li u ( ), Pl, k j and ϕ C s (), hϕ dm m()β l h s ϕ q Cs () so that h w(p k l, j ) βl 0 β l h s. Now fix ε > 0 and choose L large enough that (β 0 β 1 ) L < ε. Then h w = su l<l su h w(p k j,k 0 l, j ) + ε h s. Since there are only finitely many l, j er level l, it remains to show that the weak norm of h on k-cylinders can be made small for large k.

15 1384 M. F. Demers Choose K so that β K < ε. Then any cylinder set r Pl, r j with r K is contained in some K -cylinder K. For ϕ Cs ( r ), let ϕ r be as in 2.2 and let ϕ r denote the extension of ϕ r to K. Then β0 l hϕ dm h ϕ r dm βl 0 β K l h u ϕ Cs ( r ). r K With this choice of K and L and letting J l denote the number of l, j on level l, we have which imlies the desired comactness. h w = su su su h w(p k l<l j J l k K l, j ) + b 1 h ε, 4. Lasota Yorke-tye estimates In this section we rove Proosition 2.3. By density of Li u ( ) in B and B w, it suffices to derive these inequalities for h Li u ( ) once we show L is continuous on (B, ). To avoid reeating estimates, we ostone the roof of this fact to Weak norm estimate. Let h Li u ( ) and k 0 be fixed. Choose Pl, k j and ϕ Cs () with ϕ Cs () 1. Case 1. n l. Notice that = F n is an n + k cylinder in l n, j, i.e., P k+n l n, j. L n hϕ dm = F n hϕ F n dm m( )β n l 0 h w(p k+n l n, j ) ϕ Fn C s ( ). (14) Since J m F n 1 and d s (F n x, F n y) = α n d s (x, y) for all x, y, we have m( ) = m() and ϕ F n Cs ( ) ϕ Cs (). Putting this together with (14) and taking the suremum over ϕ Cs () and Pl, k j, yields L n h w(p k l, j ) βn 0 h w(p k+n l n, j ). (15) Case 2. n > l. First consider the case l = 0 and P0, k j. Note that F n comrises a countable union of (n + k)-cylinders, F n =, P n+k l, j. Then by (14), L n hϕ dm m( )β l 0 h w(p k+n l, j ) ϕ Fn Cs ( ) (16) F n where l, j. To estimate ϕ Fn C s ( ), take x, y γ s and write ϕ F n (x) ϕ F n (y) H s (ϕ)d s (F n x, F n y) H s (ϕ)c 0 α n d s (x, y) (17) by (2). Since ϕ F n = ϕ, we conclude ϕ F n C s ( ) C 0 ϕ C s (). Due to bounded distortion given by Proerty (P4)(b) and Lemma 2.1, we have m( )/m() C(m( n )/m( 0, j)) where n is the n-cylinder containing. Using the fact that there are only finitely many 0, j, we record the following estimate for future use. There exists C > 0 such that, m( ) Cm( n )m(). (18)

16 Now (16) becomes, L n hϕ dm F n Functional norms for Young towers 1385 CC 0 β l 0 h w m( n )m() C h w m() l, j β l 0 m( l, j ) where the sum is finite since θ < β 0. Dividing by m() and taking the suremum over ϕ C s () and P k l, j, we have L n h w(pk 0, j ) C h w. (19) Now for n > l, we use (15) and (19) to rove (6), L n h w(p k l, j ) βl 0 Ln l h w(p k+l 0, j ) C β l 0 h w Strong stable norm estimate. Fix h Li u ( ) and k 0. Choose P k l, j and ϕ C q s () with ϕ C q s () 1. Case 1. n l. Following (14) with C q s ( ) in lace of C s ( ), we obtain L n h s(p k l, j ) βn h s(p k+n l n, j ). (20) Case 2. n > l. As in 4.1, we first let l = 0 and fix P0, k j. L n hϕ dm = h(ϕ F n ϕ) dm + F n F n hϕ dm (21) where ϕ(x) = γ s (x) ϕ Fn dµ s is constant on each γ s F n. Denoting by the comonents of F n, P k+n l, j, we estimate the first term of (21) by h(ϕ F n ϕ) dm β l h s ϕ F n ϕ q Cs ( ) m( ). (22) F n F n Since ϕ F n is continuous on γ s Ɣ s and µ s is a robability measure, we have ϕ F n (z 1 ) ϕ ϕ F n (z 2 ) for some z 1, z 2 γ s. Thus using (17) we estimate ϕ F n ϕ C q s ( ) Cαqn ϕ C q s (). Putting this together with (18) and (22), we obtain h(ϕ F n ϕ) dm C β l m( n )m() h sα qn C α qn h s m(). (23) l, j F n To estimate the second term of (21), we note that ϕ Cs ( ) ϕ 1 since ϕ is constant along stable leaves. Then, again using (18) and the fact that θ < β 0, hϕ dm m( )β l 0 h w ϕ Cs ( ) C h wm() β l 0 m( l, j ). l, j F n F n (24) Combining (23) and (24), dividing by m() and taking the aroriate surema yields L n h s(p k 0, j ) Cαqn h s + C h w. (25) Now for n > l, we combine (20), (25) and the fact that α q < β to obtain (7), L n h s(p k l, j ) βl L n l h s(p k+l 0, j ) βl (Cα q(n l) h s + C h w ).

17 1386 M. F. Demers 4.3. Strong unstable norm estimate. Fix h Li u ( ) and k Pl, k j. For r k, let r Pl, r j be such that r k. For ϕ Cs ( r ), let ϕ r (x) = m( r ) 1 γ u (x) r ϕ dm γ be defined as in 2.2 and let ϕ r (x) = ϕ r (γ u (x)) be the extension of ϕ r to k. Case 1. n l. Since m( r ) = m(f n r ) and similarly for k, we have r L n hϕ dm k L n h ϕ r dm = F n r hϕ F n dm F n k h ϕ r F n dm. As before, ϕ F n Cs (F n r ) ϕ Cs ( r ). Let (ϕ Fn ) n r be the average of ϕ F n on unstable leaves in the (r + n)-cylinder r n := F n r. Since (ϕ F n ) n r is constant on unstable leaves, we may extend it to F n k P k+n l n, j. It follows that ϕ r F n = (ϕ F n ) n r since J γ F n 1 on F n k. hϕ F n dm h(ϕ F n ) n r dm h F n r F n u(p k+n l n, j ) βk+n l ϕ Cs (). k Dividing by β k l and taking the suremum over ϕ C s ( r ) and r k, we conclude that L n h u(p k l, j ) βn h u(p k+n l n, j ). (26) Case 2. n > l. As before, we first consider the case l = 0 and fix r k 0, j. Let k n denote an (n + k)-cylinder in F n k and similarly for r n n k. As usual, define (ϕ F n ) n r to be the average of ϕ F n on unstable leaves in r n, extended to n k. Now, L n h ϕ dm L n h ϕ r dm r k = m(n r ) [ h ϕ F n dm h (ϕ F n ) n k n m( F n r ) n k r k n r [ m( n + r ) m( r ) m(n k ) ] h(ϕ F n ) n m( k ) r dm + n k F n k m(n k ) k n m( F n k ) k n k n k ] dm h[(ϕ F n ) n r ϕ r F n ] dm. (27) Label the three sums of (27) by 1, 2 and 3 resectively. To estimate 1, recall that, by (17), ϕ F n Cs (r n) C 0 ϕ Cs ( r ). Let n denote the n-cylinder containing r n (and k n) on level l ( n ). We use (18) to estimate 1 Cm( n )β n+k l (n) h u ϕ F n Cs ( r n r n) C β n+k h u. (28) F n r To estimate 2, note that m(a) = m γ (γ A) for any s-subset A by Lemma 2.1(1). By Lemma 2.1(3) and (18), there exist x, y k n such that m(r n) m( r ) m(n k ) m( k ) J γ F n (y) J γ F n (x) 1 m(k n) m( k ) Cαk/2 m( n ) (29)

18 Functional norms for Young towers 1387 since s(x, y) n + k. Recall that (ϕ F n ) n r q Cs (r n) ϕ Fn q Cs (r n) C 0 ϕ q Cs ( r ) by Lemma 3.1 and (17). Using this estimate and (29), 2 Cα k/2 m( n ) h s β l C0 ϕ q Cs ( r ) C h s α k/2. (30) k n F n k In order to estimate 3, we need the following reliminary lemma. LMMA 4.1. For n k l, j, let (ϕ Fn ) n r and ϕ r be as above. There exists C > 0 deending only on the distortion of F such that (ϕ F n ) n r ϕ r F n C q s ( n k ) Cαr( q)/2 ϕ C s ( r ). Proof. For x k n, ϕ r F n (x) = m( r ) 1 ϕ dm γ = m( r ) 1 γ u (F n x) r Thus (ϕ F n ) n r (x) ϕ r F n (x) = 1 m(r n) γ u (x) n r γ u (x) n r ϕ F n J γ F n dm γ. ( ϕ F n 1 m(n r )J γ F n ) dm γ. m( r ) (31) Now m( r )/m(r n) is the average value of J γ F n on γ u (x) r n since m(r n) = m γ (γ u (x) r n) and m( r ) = m γ (γ u (F n x) r ) by Lemma 2.1(1). Lemma 2.1(3) imlies 1 m(n r )J γ F n m( r ) C 1α r/2 since the searation time for any two oints in n r is at least n + r. Thus (31) becomes (ϕ F n ) n r (x) ϕ r F n (x) ϕ C 1 α r/2. (32) It remains to estimate the Hölder constant of the difference along stable leaves. Let y γ s (x) k n. On the one hand we have ϕ r F n (x) (ϕ F n ) n r (x) ϕ r F n (y) + (ϕ F n ) n r (y) 2 ϕ C 1 α r/2, using (32) for the x and y differences searately. On the other hand, using (17), (2) and Lemma 3.1, we have ϕ r F n (x) (ϕ F n ) n r (x) ϕ r F n (y) + (ϕ F n ) n r (y) H ( ϕ r F n (γ s ))d s (F n x, F n y) + H ((ϕ F n ) n r γ s )d s (x, y) H s (ϕ)c 0 α n d s (x, y) + H s (ϕ F n )d s (x, y) 2C 0 H s (ϕ)α n d s (x, y). The Hölder constant is bounded by the minimum of the two estimates, 2C 1 α r/2 d s (x, y) q and 2C 0 α n d s (x, y) q. This minimum is largest when the two quantities are equal, i.e., when d s (x, y) = α n α r/2 C 1 /C 0. Thus H q s ( ϕ r F n (ϕ F n ) n+r ) C H s (ϕ)α qn α r( q)/2 which, together with (32), comletes the roof of the lemma.

19 1388 M. F. Demers We are now ready to estimate 3 using the strong stable norm. 3 l, j C h s β l m( l, j ) (ϕ Fn ) n r ϕ r F n C q s ( n k ) C h sα r( q)/2 (33) by Lemma 4.1. Combining (28), (30) and (33) in (27), using the fact that α ( q)/2 β, and taking the aroriate surema, we have L n h u(p k 0, j ) Cβn h u + C h s. (34) Now for n > l, we combine (26) and (34) to rove (8), L n h u(p k l, j ) βl L n l h u(p k+l 0, j ) βl (Cβ n l h u + C h s ) Continuity of L. Since L is linear, it suffices to show L is bounded on B. Note that for h B, we can exress the strong stable norm given by (4) as h s(p k l, j ) = βl su P k l, j su m() 1 h(ϕχ ) (35) ϕ q Cs () 1 where χ is the indicator function of the set. Similarly, defining ϕ r and ϕ r as in 2.2, we can exress the strong unstable norm given by (5) as h u(p k l, j ) = su su su β l k m( r ) 1 h(ϕχ r ) m( k ) 1 h( ϕ r χ k ). k Pl, k r k ϕ j Cs (r ) 1 (36) Now following the estimates of 4.1, we have for Pl, k j and ϕ Cq s (), Lh(ϕχ ) = h(ϕ F χ ) m( )β l ( ) h s ϕ F q Cs ( ). (37) F 1 F 1 By (17), ϕ F C q s ( ) C 0 ϕ C q s (). Now if l > 0, then there is only one = F 1, l ( ) = l 1, and m( ) = m() so we conclude that Lh s(p k l, j ) C 0β h s. On the other hand, if l = 0, then by (18), m( ) Cm( 1 )m() where 1 is the 1-cylinder containing. So (37) becomes Lh(ϕχ ) CC 0 m() h s β l m( l, j ) F 1 and the sum is finite since β > θ. Thus Lh s C h s as required. Similarly, using (36) and following the estimates of 4.3, one sees that Lh u C( h s + h u ). 5. Sectral icture Proosition 2.3 and Lemma 3.6 imly that L : B is quasi-comact with essential sectral radius bounded by β. In this section we study the eriheral sectrum of L on B and rove Theorems 1 and 2.

20 5.1. Proof of Theorem 1. Functional norms for Young towers 1389 LMMA 5.1. The sectral radius of L on B is 1. measures and contains no Jordan blocks. Its eriheral sectrum comrises Proof. It is a direct consequence of (6) that the sectral radius of L is at most one. If it were strictly less than one, Lemma 3.2 would yield the contradiction m(1) = L n m(1) = lim n Ln m(1) lim n C Ln m = 0. Now let z be in the sectrum of L with z = 1. Suose there exist h 0, h 1 B, h 0 0 such that Lh 0 = zh 0 and Lh 1 = zh 1 + h 0. Then L n h 1 = z n h 1 + nz n 1 h 0. Now (17) imlies that Hs (ϕ F n ) C 0 α n Hs (ϕ) for ϕ Cs ( ). So n h 0 (ϕ) h 1 (ϕ) + L n h 1 (ϕ) = h 1 (ϕ) + h 1 (ϕ F n ) 2C h 1 w ( ϕ + C 0 H s (ϕ)) for each n by Lemma 3.2. Dividing by n and taking the limit as n imlies that h 0 0, contrary to our assumtion. It remains to show that the eriheral sectrum is comrised of measures. Suose h B satisfies Lh = zh for some z = 1. By Lemma 3.2 and (17), h(ϕ) = L n h(ϕ) = h(ϕ F n ) C h w ( ϕ + C 0 α n H s (ϕ)). Letting n yields h(ϕ) C h w ϕ for all ϕ Cs ( ), which imlies h is a measure. Let V φ denote the eigensace in B corresonding to the eigenvalue e iφ. The absence of Jordan blocks in the eriheral sectrum imlies that the sectral rojectors φ : B V φ are well-defined and satisfy 1 n 1 φ h = lim e ikφ L k h (38) n n where convergence is in the -norm so the limit holds with h alied to any ϕ C s ( ). By density, φ Li u ( ) = V φ so that for each η V φ there is an h Li u ( ) such that φ h = η. LMMA 5.2. Let η V φ. Then η is absolutely continuous with resect to ν where ν := 0 1. Moreover, η has absolutely continuous conditional measures with resect to Riemannian volume on unstable curves γ Ɣ u ( ) and η G. Proof. Let h η Li u ( ) be such that φ h η = η. Then for ϕ C s ( ), 1 n 1 η(ϕ) lim e ikφ L k h η (ϕ) lim n n h 1 n 1 η L k 1( ϕ ) = h η ν( ϕ ). n n k=0 So η = ψν for some ψ with ψ L (ν) h η. Since η can be written as the limit of η n = (1/n) n 1 k=0 e ikφ L k h η and the conditional measures of η n on unstable leaves in Ɣ u have uniformly bounded and Lischitz densities, the Markov structure of and the regularity of J u F guarantee that this roerty asses to η (this can be roved as in [Y1, 2]). k=0 k=0

21 1390 M. F. Demers LMMA 5.3. The eriheral sectrum of L on B forms a grou on the unit circle. Proof. Let U F ϕ = ϕ F define the comosition oerator of F on L 2 (ν) and let L ν be its dual, i.e., the transfer oerator of F with resect to ν. It is a standard fact of ergodic theory that the eigenvalues of U F form a subgrou on the unit circle [W, 3.1]. It is then straightforward to show that the the eriheral eigenvalues of L ν equal those of U F. In what follows, we will show that the eigenvalues on the unit circle of L ν acting on L 2 (ν) are recisely the eriheral sectrum of L on B. We denote these sets by ϱ(l ν ) and ϱ(l) resectively. Suose η V φ and let ψ L (ν) satisfy η = ψν by Lemma 5.2. For ϕ Cb 0( ), e iφ ϕψ dν = e iφ η(ϕ) = Lη(ϕ) = η(ϕ F) = ϕ F ψ dν = ϕ L ν ψ dν so that L ν ψ = e iφ ψ. Since ψ L (ν) L 2 (ν), we conclude that ϱ(l) ϱ(l ν ). To show inclusion in the other direction, suose ψ L 2 (ν) satisfies L ν ψ = e iφ ψ. For j 0, choose h j Li u ( ) Cs ( ) such that ψ h j L 1 (ν) 1/j. By Lemma 3.3(ii), h j ν B. Thus, 1 n 1 φ (h j ν) = lim e ikφ L k (h j ν) =: η j =: ψ j ν V φ, (39) n n k=0 by (38) and Lemma 5.2. On the other hand, for ϕ C 0 b ( ), 1 n 1 e ikφ L k (h j ν)(ϕ) = 1 n 1 e ikφ n n k=0 k=0 The first term in (40) is bounded by k=0 ϕ F k (h j ψ) dν + 1 n 1 ϕ ψ h j n L 1 (ν) ϕ /j. ϕψ dν. (40) Combining (39) and (40), we have ϕψ dν ϕψ j dν ϕ /j for all ϕ Cb 0( ) and each j. Since V φ is finite dimensional and ψν is aroximated by elements of V φ, it must be that ψν V φ. Thus ϱ(l ν ) ϱ(l). We call 0, j a recurrent base if µ-almost every x 0, j satisfies: F n (x) 0, j for infinitely many n > 0. We call 0, j transient if it is not recurrent. LMMA 5.4. Let ( j) be the tower above a recurrent base 0, j. Then the full Lebesgue measure of ( j) belongs to a single ergodic comonent. Proof. Choose ν V 0 ergodic and fix a density oint x 0 which is not on the boundary of any cylinder P k l, j, k 0. Without loss of generality, take x 0 0, j, a recurrent base. Let γ 0 be the unstable leaf containing x 0 and let ρ 0 be the conditional density of ν on γ 0 which is Lischitz by Lemma 5.2. Thus ρ 0 > 0 on an oen subset U in γ 0 containing x 0. Since 0, j is recurrent, there exists P k 1, j with γ 0 U such that F n () is a u-subset of 0, j for some n > 0. Thus the density of F n ν F n ( γ 0 ) is strictly ositive. Since each stable leaf γ s Ɣ s ( ) belongs to a single ergodic comonent, the full Lebesgue

22 Functional norms for Young towers 1391 measure of 0, j must belong to a single ergodic comonent. Thus the ergodic comonent of ν includes the full measure of each recurrent base (and hence the entire tower above each such base) which contains a density oint of ν. Proof of Theorem 1. Lemma 5.4 allows us to assign each of the recurrent bases to a single ergodic comonent: two bases 0, j and 0, j are in the same ergodic comonent if and only if there exists n, n such that F n ( 0, j ) 0, j and F n ( 0, j ) 0, j. Ste 1. Mixing case. By transitivity the full Lebesgue measure of each base belongs to a single ergodic comonent by Lemma 5.4; therefore there can be only one invariant robability measure ν V 0. Now suose there exists η V φ where φ = 2π /q for some, q Z +. Thus ν and η are both invariant densities for L q. Since F is mixing, F q is also transitive on so L q can have at most one invariant robability measure. Thus η = ν. This roves item (i) of the theorem. Ste 2. Non-mixing case. First assume F is transitive and eriodic with eriod. Then decomoses under F into transitive comonents, on each of which F is mixing. By Ste 1, 1 is an eigenvalue of L with multilicity and there are no other eigenvalues on the unit circle. The corresonding eigenvalues for L lie at the th roots of unity and it follows from transitivity that all the th roots are realized as simle eigenvalues. This roves (ii). If F is not transitive, we simly restrict to a single transitive comonent and aly (ii). Since quasi-comactness imlies there are only finitely many comonents, (iii) follows Proof of Theorem 2. (i) Theorem 2(i) is roven by Lemma 5.2. (ii) Let ν be a hysical measure. There exists B ν with µ(b ν ) > 0 such that for every ϕ Cb 0, 1 n 1 lim ϕ(f i x) = ν(ϕ) for all x B ν. (41) n n i=0 Let x 0 l, j be a density oint of B ν. Given ε > 0, let h ε Li u ( ) be a robability density with resect to m suorted on l, j such that h ε (B ν ) 1 ε. For ϕ C 0 b ( ), choose a set B ν,ε B ν on which (41) converges uniformly and h ε (B ν \B ν,ε ) 1 2ε. Then 1 n 1 0 h ε (ϕ) = lim h ε (ϕ F i ) n n i=0 1 n 1 = lim h ε (ϕ F i 1 Bν,ε ) + h ε (ϕ F i 1 l, n n j \B ν,ε ) i=0 = h ε (1 Bν,ε )ν(ϕ) + O( ϕ ε) = ν(ϕ) + O( ϕ ε). Since 0 h ε V 0, this means ν can be aroximated by elements of V 0 and so ν V 0. The fact that ν is ergodic follows from its definition as a hysical measure.

23 1392 M. F. Demers Next we show that all ergodic elements of V 0 are hysical measures. By Lemma 5.4, if ν V 0 is an ergodic measure, its suort is contained in a finite set of towers ( j) above recurrent bases 0, j and no other ergodic measure in V 0 can be suorted on this set. Denote this set by ν and note that ν( ν ) = 1. Thus (41) converges to ν(ϕ) for ν- almost every x ν. Since the ergodic average is constant along stable leaves and ν has absolutely continuous conditional measures on γ Ɣ u ( ), we must have (41) converging for µ γ -almost every x γ on every γ Ɣ u ( l, j ), l, j ν. On the other hand, if 0, j is transient, every oint in ( j) belongs to a cylinder set of finite length n which mas to a recurrent base 0,k at time n. Let C k denote the collection of such cylinder sets in ( j) which ma to the recurrent base 0,k. It is clear that C k belongs to the basin of attraction of the unique element of V 0 suorted on 0,k and that the forward ergodic averages beginning in C k converge to the same constant. Thus to each ergodic element ν V 0, we associate a basin B ν comrising a maximal set of recurrent towers ν lus a collection of cylinder sets from transient towers. Since (41) converges for µ-almost every x B ν and ν(b ν ) = ν( ν ) = 1, ν is necessarily a hysical measure. It remains to show that the ergodic decomosition with resect to ν corresonds to that with resect to m. Let ν be an ergodic element of V 0. Since F 1 B ν = B ν and B ν is a union of cylinder sets, 1 Bν C s ( ) and 1 Bν F = 1 Bν, so 1 n 1 ν(b ν ) = lim L i 1 n 1 m(b ν ) = lim m(b ν ) = m(b ν ). n n n n i=0 (iii) This is imlied by the argument in (ii). (iv) This is a standard corollary of the existence of a sectral ga given by Theorem 1(i). To obtain the slightly stronger result for ψ Li u ( ) and ϕ C s ( ), note that ψν G and thus ψν B by Lemmas 5.2 and 3.3(i). Thus there exists σ < 1 such that ψϕ F n dν = L n (ψν)(ϕ) = ν(ψ)ν(ϕ) + O(σ n ψν B ϕ C s ( ) ). i=0 6. Large-deviation estimates We connect the moment generating function q(z) defined in to the sectral roerties of a generalized transfer oerator as follows. For g Li u ( ) C s ( ), define the generalized transfer oerator for h B by L g h(ϕ) = h(ϕ F e g ) for all ϕ C s ( ). It is then a simle calculation that L n g h(ϕ) = h(ϕ Fn e S ng ). Now fix g and for z C suose that L z = L zg is quasi-comact with a simle eigenvalue λ z of maximum modulus. Then, for h B such that λz h(1) 0, 1 q(z) = lim n n log h(ezsng 1 ) = lim n n log Ln z h(1) = log λ z. (42) So q(z) is well-defined if L z has a sectral decomosition similar to that for L 0 = L given by Theorem 1. With this in mind, we rove in 6.2.