MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS

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1 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT Abstract. We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville Pomeau map.. Introduction and Notation In this paper we look at the multifractal analysis of some non-uniformly hyperbolic maps. In particular we look at the problem for the Birkhoff averages of continuous functions. This type of problem is well understood in the hyperbolic case (see [],[6],[8] for specific results and [7] for an introduction to the subject). However in the non-uniformly hyperbolic case much less is known. The results known so far concerning Hausdorff dimension of such spectra are limited to Lyapunov spectra ([3],[9],[5]) and local dimension of Gibbs measures ([2]). Furthermore the methods cannot be applied to Birkhoff averages for general continuous functions. In the case of general continuous functions there are results for topological entropy [2], but not for Hausdorff dimension. In this paper we produce results for the Hausdorff dimension which extend some of the results of [6] into the non-uniformly hyperbolic setting. We begin with a classical example. Let T : [0, ] [0, ] be the Manneville Pomeau map defined by T x = x + x +β mod, where 0 < β <. Let f : [0, ] R be continuous and define Λ α = { n x [0, ] : lim f(t i x) = α n n Let us denote, α min = inf µ E { f dµ and α max = sup µ E { f dµ, where E = E T ([0, ]) denotes the space of T -invariant ergodic probabilities. i= Mathematics Subject Classification. 37C45 (37D25 28A80). Key words and phrases. multifractal formalism, dimension theory, dynamical system.

2 2 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT For α [α min, α max ]\{f(0) we have { h(µ, T ) dim H Λ α = log T (x) dµ : sup µ M T ([0,]) f dµ = α, where M T ([0, ]) denotes the T -invariant probability measures. We can also show that dim H Λ f(0) =. We can consider the related problem for iterated function systems. Let T i : [0, ] [0, ], i m, be C maps such that T i (x) > 0 and for distinct i, j, we have T i (0, ) T j (0, ) =. At this stage the only additional assumption we make is that diam(t i T in ([0, ])) converges to 0, uniformly in all sequences of maps. Let A = {,..., m and let Σ = A N be the one-sided shift space on m symbols, σ : Σ Σ the usual shift map and let f : Σ R be a continuous function. Given n we let A n f(ω) = n n i=0 f(σi ω) denotes the nth level Birkhoff average of the function f : Σ R. Let Π : Σ [0, ] be the natural projection defined by Π(ω) = lim n T ω T ωn (0), ω Σ. Furthermore we can define the attractor of the system by Λ = Π(Σ). We will consider the sets { X α = ω Σ : lim A nf(ω) = α n and their images Π(X α ) Λ. Let M σ (Σ) denote the σ-invariant measures and let E σ (Σ) denote the σ-ergodic measures we will use M σ n(σ), and E σ n(σ) to denote the σ n -invariant and σ n -ergodic measures respectively. For µ M σ n(σ) let h(µ, σ n ) denote the entropy of µ with respect to σ n. Let F n be the finite algebra generated by the n-cylinders. Since, for each n, F n is a generating partition for σ n, we have for µ M σ n(σ) that h(µ, σ n ) = nh(µ, σ) where and H(ν) denotes the Shannon entropy h(µ, σ) = lim N N H (µ F N ) H(ν) = A A ν(a) log(/ν(a)), defined for probabilities ν on finite algebras A. Let g : Σ R be defined by g(ω) = log T i (Π(σ(ω))). Lyapunov exponent of a measure µ M σ n(σ) by λ(µ, σ n ) = n g(σ k ω) dµ(ω). k=0 We then denote the

3 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 3 Let Σ be the subset of Σ on which the diameters tend to 0 exponentially, i.e. let Σ = {ω : lim inf n A ng(ω) > 0. We denote, α min = inf µ Eσ(Σ){ f dµ and α max = sup µ Eσ(Σ){ f dµ. We can now state our first result as follows. Theorem. For α [α min, α max ] we have dim H Π((X α ) Σ) = sup µ M σ(σ) { h(µ, σ) λ(µ, σ) : f dµ = α and λ(µ, σ) > 0. We now consider the specific case where T,..., T m : [0, ] [0, ] are C maps with fixed points (x,..., x m ) such that T i (x i ) = x i. We assume that T i (x i) and 0 < T i (x) < everywhere else, and for distinct i, j, we have T i(0, ) T j (0, ) =. Furthermore we assume that for any ɛ > 0 there exists a µ M σ (Σ) with λ(µ, σ) > 0 and h(µ,σ) λ(µ,σ) dim H Λ ɛ. In other words we have a system with a finite number of parabolic fixed points with hyperbolic measures with dimension arbitrarily close to that of the attractor. If the maps T i are all C +s for some s > 0 then we can deduce from Theorem 4.6 in [0] that this condition is satisfied. We can use Theorem to deal with the cases where the Lypaunov exponent is nonzero. We use a method similar to the work of Gelfert and Rams [3] to deal with the cases where the Lyapunov exponent can be zero. Let ω i = (i, i, i,... ), x i = Π(ω i ) and α i = f(i, i, i,...). For i such that T i (x i) =, let (.) A = [min{α i, max{α i ]. i i Theorem 2. Assume that the iterated function system has a finite number of indifferent fixed points as above and that for any ɛ > 0 there exists a measure µ M σ (Σ) with λ(µ, σ) > 0 and h(µ,σ) λ(µ,σ) dim H Λ ɛ. Then, for α [α min, α max ] \ A we have { h(µ, σ) dim H Π(X α ) = λ(µ, σ) : f dµ = α sup µ M σ(σ) and dim H Π(X α ) = dim H Λ for all α A. It is straightforward to deduce that in this case dim H Π(X α ) is a continuous function of α with the possible exception of the endpoints of A. Corollary. The function r : [α min, α max ] R defined by r(α) = dim H Π(X α ) is constant in the interior of A and continuous in [α min, α max ]\A. Proof. Since r is clearly constant in the interior of A we just consider α [α min, α max ]\ A. To start let µ, µ 2 M σ (Σ) such that f dµ = α min and f dµ 2 = α max. For

4 4 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT α [α min, α max ]\A consider a sequence {α n n N such that α n α as n. It follows that r(α) lim sup n r(α n ) by upper semicontinuity of entropy (see Theorem 8.2 []). For ɛ > 0 we let µ satisfy f dµ = α and h(µ,σ) λ(µ,σ) > r(α) ɛ. By considering the measures pµ + ( p)µ and pµ 2 + ( p)µ for appropriate p sufficiently small it is clear that r(α) lim inf n r(α n ). These results are well understood in the case of uniformly contracting systems (see [8],[6],[]). The novelty in this work is that we can analyse certain non-uniformly hyperbolic systems. Moreover, our methods do not involve either thermodynamic formalism or the use of large deviation theory. For an introduction to dimension theory and multifractal analysis the reader is referred to [7]. All the necessary definitions and results from ergodic theory can be found in []. We can also use Theorem 2 to deduce a result which applies to non-uniformly expanding maps of the interval (such as the Manneville Pomeau map mentioned earlier). Corollary 2. Let T : [0, ] [0, ] be a piecewise C Markov map with a finite number of parabolic fixed points x i such that T (x i ) = x i and T (x i ) = but T (x) > for x [0, ]\ i x i. We also assume the existence of a hyperbolic measure with dimension arbitrarily close to. Let f : [0, ] R be continuous and let n Λ α = {x [0, ] : lim f(t i x) = α. n n If we let A = [min i {f(x i ), max i {f(x i )] then for α [α min, α max ]\A we have { h(µ, T ) dim H Λ α = log T (x) dµ(x) : f dµ = α. sup µ M T ([0,]) We also have that for all α A i=0 dim H Λ α =. Proof. This follows by noting that Theorem 2 can be applied to the iterated function system defined by the inverse branches of this map. Without the assumption of the existence of a hyperbolic measure of dimension arbitrarily close to the result would be the same except that we would no longer have equality for α A but that the dimension is bigger than the supremum of the dimension of hyperbolic measures and less than the dimension of the attractor. We don t know of any examples where this situation occurs. It follows form Theorem 4.6 in [0] that for any such system where the inverse branches are C +s for s > 0 that

5 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 5 this hypothesis is satisfied. We now give some examples to illustrate this corollary and the difference of the result from the expanding case. Example. The Manneville Pomeau map is known to have a finite T -invariant absolutely continuous probability measure (we denote this µ SRB ) in the case when 0 < β <. For β there is no T -invariant absolutely continuous probability measure but there are measures of dimension arbitrarily close to. So provided β > 0 it satisfies the hypotheses of the Corollary 2. Let f : [0, ] R be a continuous function such that f dµ SRB > f(0) = α min. Then for β (0, ) we have that { = for α [ f(0), ] f dµ SRB dim H Λ α < for α [ ] f dµ SRB, α max In the case where β > we have that dim H Λ f(0) = and dim H Λ α < for α (f(0), α max ). Example 2. Let T : [0, ] [0, ] be defined by { x T (x) = x for 0 x 2 2x x for 2 < x. Thus T has parabolic fixed points at 0 and but is expanding everywhere else. There are no absolutely continuous T -invariant probability measures but there are T -invariant measures of dimension arbitrarily close to. Hence for any f : [0, ] R which is continuous we can apply Corollary 2. In the case where f is monotone increasing then A = [f(0), f()] = [α min, α max ] and dim H Λ α = for all α [α min, α max ]. The layout of the rest of the paper is as follows. In the next three sections we give the proof of Theorem. In the final section we go on to deduce Theorem Preliminary Results In this section we prove the basic lemmas needed to prove Theorem. These involve an approximation result and methods of relating measures in M σ n(σ) and M σ (Σ) to those in E σ (Σ). In the following, let π n : Σ A n denote the natural projection onto the first n symbols, i.e. π n (ω) (ω,..., ω n ) A n. For cylinder sets we use the notation [α,..., α n ] for πn (α,..., α n ) and for ω Σ we let [ω] n denote (π n (ω)). Let F n be the finite σ-algebra generated by the n-cylinders {[ω] n : ω π n Σ.

6 6 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT For a function f : Σ R, define the nth variation as var n f = sup [ω ] n=[ω] n f(ω) f(ω ). By definition, lim n var n f = 0 if f is continuous and then also lim n var n A n f = 0. Let I n (ω) I denote the interval T ω T ωn ([0, ]) and let the corresponding diameters be given by D n (ω) = diam(i n (ω)). We write λ n (ω) for n log D n(ω). We start by showing that λ n is well approximated for large n by the Birkhoff average A n g(ω). Lemma. Let T i : [0, ] [0, ], i m, be C maps such that T i (x) > 0 and such that for distinct i, j, we have T i (0, ) T j (0, ) =. In addition we assume that D n (ω) 0, uniformly in ω. Then lim sup { n n log D n(ω) A n g(ω) = 0 ω Σ Proof. We introduce the functions g n : Σ R, n, defined by g n (ω) = log It is immediate from the definitions that D n (ω) D n (σω). n log D n (ω) = g n i (σ i ω). i=0 We can relate this identity to the Birkhoff averages of g : Σ R using the following fact (2.) g n (ω) g(ω) uniformly in ω as n. To see (2.), we note that for n 2, ( g n (ω) = log D n (σω) I n (σω) T ω (x) dx ) = log T ω (ξ) for some ξ I n (σω) by the intermediate value theorem. By hypothesis, each log T i (x) is (uniformly) continuous and thus, since diam(i n (σω)) = D n (σω) tends to 0 uniformly, it follows that also tends to 0 uniformly as n. g n (ω) g(ω) = log T ω (Π(σω)) log T ω (ξ) Let ɛ > 0 and note that by (2.), we can choose N such that for n N and ω Σ we have g n (ω) g(ω) ɛ 2. We can also find a C > 0 where g n (ω) g(ω) < C for all ω Σ. Let N = ([ ] ) 2C ɛ + N.

7 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 7 For n N and ω Σ we have that n log D n (ω) = g n i (σ i ω) The other inequality is similar. i=0 n N i=0 ( g(σ i ω ) + ɛ n 2 ) + i=n N g n i (σ i ω) n g(σ i ω) + (n N ) ɛ 2 + C N i=0 n g(σ i ω) + (n N) ɛ 2 + N ɛ 2 i=0 n ( g(σ i (ω)) + ɛ ). i=0 We now need to relate σ n -ergodic measures to σ-ergodic measures and σ-invariant measures to σ n -ergodic measures. Given ν E σ n(σ) we define µ = A nν as the measure µ = n ν σ k. n k=0 Lemma 2. If µ = A nν, ν E σ n(σ), then µ E σ (Σ) and () h(µ, σ) = n h(ν, σn ). (2) λ(µ, σ) = n λ(ν, σn ). (3) f dµ = A n f dν. Proof. Let A Σ be a measurable set. Then since ν(σ n A) = ν(a) we have n n µ(σ A) = ν σ (k+) (A) = ν σ k (A) = µ(a) k=0 and deduce that µ is σ-invariant. For any σ-invariant set A = σ (A), we have µ(a) = n n k=0 ν(σ k A) = ν(a) and since A is also σ n -invariant and ν is σ n -ergodic, we conclude that µ(a) = ν(a) is either 0 or. k=0 Since ν is σ n -invariant and µ is σ-invariant, we know that f dµ = A n f dµ = A n f dν.

8 8 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT We can show in an identical fashion that λ(µ, σ) = n λ(ν, σn ). We can see that h(µ, σ) = n h(ν, σn ) by Abramov s Theorem (see [] Theorem 4.3). The next lemma states that we may approximate any invariant measure µ M σ (Σ) by ergodic measures in E σ n(σ). A probability measure µ on Σ is nth level Bernoulli if the n-blocks π n T kn (ω) are independent and identically distributed for k = 0,,.... An nth level Bernoulli measure is always σ n -invariant and ergodic with respect to σ n. Moreover, we have a natural continuous bijection ν ν between probabilities on blocks ν M(A n ) and the corresponding nth level Bernoulli measures. The block probability ν is the marginal of the corresponding nth level Bernoulli measure, i.e. (ν ) πn = ν, and we have h(ν, σ n ) = H(ν). Lemma 3. For any µ M σ (Σ), we can find a sequence of measures {µ n converging to µ in the weak topology such that () µ n is nth level Bernoulli, (2) lim n n h(µ n, σ n ) = h(µ, σ) and lim n n λ(µ n, σ n ) = λ(µ, σ). Moreover, if f dµ = α (α min, α max ) then we may in addition assume that (3) A n f dµ n = α, Proof of Lemma 3. To see the first part of the lemma let µ n = (µ πn ). Then µ n Fn = µ Fn and, since F n increases to the Borel σ-algebra, this implies that µ n µ in the weak topology and also that λ(µ, σ) = lim A n g dµ n and α = n and, by definition, that f dµ = lim n h(µ, σ) = lim H(µ π n n ) = lim n n + n h(µ n, σ n ). A n f dµ n We need to work a little bit more to modify this construction to give a sequence µ n of nth level Bernoulli measures that also satisfies (3), in addition to () and (2). Without loss of generality, we may assume that A nj f dµ nj α for some infinite sequence N = {n j. We proceed to construct µ n for such n s and a symmetric construction gives µ n for n N \ N. By the ergodic theorem, we can always find a point x Σ, a number 0 < ρ < (α max α)/3 and an integer N > 0 such that A n f(x) α+3ρ for all n N. Denote by ν n = µ n πn M(A n ) the block marginal of µ n and let δ n M(A n ) denote the Dirac measure at the word (x, x 2,..., x n ). Then δn is a Dirac measure on the

9 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 9 corresponding periodic sequence, and we can also assume that A n f d(δ n ) > α + 2ρ for all n N. Furthermore, define for s [0, ] a nth level Bernoulli measure ξ s,n := (sν n + ( s)δ n ). Note that the map s F (s) = A n f dξ s,n is continuous and hence, since n N means that F () α and F (0) > α + 2ρ, we deduce that A n f dξ sn,n = α, for some s n [0, ]. We need to show that we can choose s n as n tends to infinity with n N, since then by setting µ n = ξ s,n we have and n h( µ n, σ n ) = s n n h(µ n, σ n ) + ( s n ) 0 + O(/n) h(µ, σ) λ( µ n, σ) = s n λ(µ n, σ) + ( s n ) A n g d(δ n ) + var n A n g λ(µ, σ) as required. For all s [0, ], the n-block marginals for ξ s,n and ζ s,n := sν n + ( s)δ n coincide, i.e. ξ s,n πn = ζ s,n πn. Therefore A n f dξ s,n A n f dζ s,n var n A n f. and since νn = µ n we deduce A n f dζ s,n = s A n f dµ n + ( s) A n f d(δ n ) α sɛ n + ( s) (2ρ), where ɛ n = α A n f dµ n. Thus α = A n f d µ n α sɛ n + ( s) (2ρ) var n A n f, which, since ɛ n 0 and var n A n f 0 as n N tends to infinity, gives that lim n + s n =.

10 0 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT 3. Lower Bound In this section we shall prove the lower bound in Theorem, i.e. { (3.) dim H Π(X α Σ) h(µ, σ) sup µ M σ(σ) λ(µ, σ) : f dµ = α, λ(µ, σ) > 0. We start by calculating the dimension of the projection of any invariant measure µ E σ n(σ) with positive Lypaunov exponent. Lemma 4 (Hofbauer Raith). Let µ M σ n(σ) be ergodic with respect to σ n and satisfy λ(µ, σ n ) > 0. We have that dim H (µ Π ) = h(µ, σn ) λ(µ, σ n ). Proof. This was originally shown by Hofbauer and Raith in [4]. The proof can be seen by applying Lemma, together with the Birkhoff Ergodic Theorem and the Shannon McMillan Breiman Theorem. Let µ E σ n(σ) satisfy both A n f dµ = α and λ(µ, σ n ) > 0. It follows from the Birkhoff Ergodic Theorem that for any such µ we have µ(x α Σ) =. Hence we can deduce from Lemma 4 that and so dim H Π(X α Σ) dim H Π(X α Σ) h(µ, σn ) λ(µ, σ n ) { h(µ, σ n ) sup µ E σ n(σ) λ(µ, σ n ) : We can now apply Lemma 3 to see that dim H Π(X α Σ) for α (α min, α max ). sup µ M σ(σ) { h(µ, σ) λ(µ, σ) : A n f dµ = α and λ(µ, σ n ) > 0. f dµ = α and λ(µ, σ) > 0 The cases when α = α min or α = α max need to be handled separately since we cannot apply Lemma 3. However it can be seen from the ergodic decomposition of such an invariant measure that must be the same as sup sup { h(µ, σ) λ(µ, σ) : µ M σ(σ), { h(µ, σ) λ(µ, σ) : µ E σ(σ), f dµ = α min f dµ = α min.

11 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS If µ E σ (Σ) occurs in an ergodic decomposition of µ M σ (Σ) where f dµ = α min, then, by the extremality of α min, f dµ = α min. The same is true for α max. This completes the proof of the lower bound. 4. Upper Bound In this section we shall prove the upper bound in Theorem, i.e. { (4.) dim H Π(X α Σ) h(µ, σ) sup µ M σ(σ) λ(µ, σ) : f dµ = α, λ(µ, σ) > 0. For δ > 0 let Σ(δ) = {ω Σ : lim inf n A ng(ω) δ. so that Σ can be written as the countable union Σ = δj Σ(δj ), for any sequence {δ j with lim j δ j = 0. Recall that D n (ω) = diam (T ω T ωn ([0, ])) and λ n (ω) = n log D n(ω). An important consequence of Lemma is that for any η > 0 there exists some N 0 = N 0 (η) such that (4.2) λn (ω) ( η)a n g(ω) for all n N 0 and all ω Σ(δ). (To see this, take ɛ = ηδ in the proof of Lemma.) Most of the section will be devoted to proving the following lemma. We consider the sets { X(α, N, ρ, δ) = ω Σ(δ) : A n f(ω) B(α, ρ) for all n N. Lemma 5. For all ρ, ɛ > 0, δ > 0 sufficiently small and N N we can find a measure µ M σ (Σ) such that f dµ B(α, 2ρ), λ(µ, σ) > δ and dim H Π(X(α, N, ρ, δ)) h(µ, σ) λ(µ, σ) + ɛ. Before proving this lemma we show how it can be used to deduce (4.). Proof of (4.). First note that X α Σ(δ) N N X(α, N, ρ, δ). for any ρ, δ > 0. Thus, for any fixed ρ, δ, we have dim H Π(X α ) sup dim H Π(X(α, N, ρ, δ)). N

12 2 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT For any ɛ, δ > 0 and any sequence ρ n, decreasing to zero as n, we can use Lemma 5 to find a sequence of invariant measures µ n M σ (Σ) with f dµ n B(α, 2ρ n ), λ(µ n, σ) > δ and dim H Π(X α ) dim H Π(X(α, N n, ρ n )) + ɛ/2 h(µ n, σ) λ(µ n, σ) + ɛ. If µ δ is any weak -limit of µ n it clearly follows that f dµ δ = α, and from the upper semicontinuity (see [] Theorem 8.2) of entropy µ h(µ, σ) and the continuity of µ λ(µ, σ) that dim H Π(X α Σ(δ)) h(µ δ, σ) λ(µ δ, σ) + ɛ, where λ(µ δ, σ) > δ. This holds for any δ > 0 so by taking a countable union of δ n where δ n 0 we get { dim H Π(X α Σ) h(µδn, σ) sup n λ(µ δn, σ) + ɛ. Since ɛ was arbitrary, this completes the proof of (4.). Proof of Lemma 5. Fix ɛ > 0, N N, α, δ, ρ > 0 and let X = X(α, N, ρ, δ). By compactness, f and g are uniformly continuous and if N is sufficiently large then for all n N, (4.3) A n f(τ) A n f(ω) ρ/2 whenever [ω] n = [τ] n. Furthermore, by (4.2) we can assume that for n N, (4.4) A n g(ω) ( + ɛ) λ n (ω). For n N, let Y n consist of all cylinders [ω] n F n which contain a point in X, i.e. Y n = π n (X). For each n, define s n to be the solution to (4.5) n (ω) =. [ω] n Y n D sn From the definition of Hausdorff dimension, it then immediately follows that dim H Π(X) lim inf n s n, since the projections of the elements of Y n form a sequence of covers of Π(X) by intervals having diameters decreasing to zero as n. Let ν n be the probability defined by (4.5) on the n-cylinders, i.e. ν n ([ω] n ) = D sn n (ω) = e nsn λ n(ω)

13 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 3 if [ω] Y n and zero otherwise. Let µ n be the corresponding nth level Bernoulli measure defined by ν n = µ n Fn. That is, µ n = νn where ν n is interpreted as a measure in M(A n ). It is clear from (4.3) that A n f dµ n B(α, 2ρ), since each cylinder [ω] n Y n only contains τ Σ such that A n f(τ) B(α, 2ρ). Evaluating the Shannon entropy H(ν n ) of ν n gives the equality H(ν n ) = ns n λ n dν n, where the integral denotes the expected value [ω] n ν([ω] n ) λ n (ω) of λ n with respect to ν n. Since µ n = ν n and λ n is F n -measurable, it is easy to see that H(ν n ) = h(µ n, σ) and that λn dµ n = λn dν n. Thus we have the identity h(µ n, σ) = s n λ n dµ n. In view of (4.4), this means that (4.6) h(µ n, σ n ) s n ( ɛ) A n g dµ n = ( ɛ)s n λ(µ n, σ n ) since µ n is nth level Bernoulli and σ n -invariant. Thus (4.7) s n h(µ n, σ n ) ( ɛ) λ(µ n, σ n ) To complete the proof of Lemma 5, simply note that Lemma 2 implies that A nµ n belongs to E σ (Σ) and that h(a nµ n, σ) = n h(µ n, σ n ) and that λ(a nµ n, σ) = n λ(µ n, σ n ). Moreover, A n f dν n B(α, 2ρ). 5. Proof of Theorem 2 We now need to consider the sequences ω where lim inf n A n g(ω) = 0 and we cannot apply Theorem. We start by showing that if the limit of the Birkhoff average for such a sequence exists it must necessarily lie in A (see (.) for the definition). Lemma 6. Let {n j j N be a subsequence of N. If for any ω Σ, lim j A nj g(ω) = 0, then we have that lim j A nj f(ω) A if the limit exists. In particular, this shows that lim n A n g(ω) = 0 means lim n A n f(ω) A, if the limit exists. Proof. Let ɛ, l > 0. We can find δ > 0 such that g(ω) < δ implies that dist(f(ω), A) < ɛ 2. If n j A nj g(ω) < δ l then g(σ i ω) < δ for at least n j(l ) l of the values i n j.

14 4 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT Thus dist(a nj f(ω), A) (l ) l ɛ + K l. Where k = max ω Σ{dist(ω, A). Since l is arbitrary this completes the proof. Hence we know that when α / A, we do not need to consider those ω which satisfy lim inf n A n g(ω) = 0, since then, if lim n A n f(ω) exists, it can only take the values in A. Thus the first part of Theorem 2 follows immediately from Theorem. For the second part for convenience we will write A = [A, A 2 ] and by renaming symbols we may arrange so that for k =, 2, T k (x k ) = x k, x k = Π(ω k ), f(ω k ) = A k and ω k = (k, k, k,... ). We will denote by δ and δ 2 the Dirac measures on the sequences ω and ω 2, respectively. We will need to consider separately the two cases where α (A, A 2 ) and when α {A, A 2. For the first case we need the following. Lemma 7. For any α (A, A 2 ) and ν E σ (Σ) such that λ(ν, σ) > 0 we can find µ M σ (Σ) such that f dµ = α and h(µ, σ) h(ν, σ) = λ(µ, σ) λ(ν, σ). Proof. We fix α (A, A 2 ), ν E σ (Σ) and assume for convenience that f dν α. We can then find p, p 2 > 0 such that p + p 2 =, p 2 > 0 and p A 0 + p 2 f dµ = α. We then let We then have that µ = p δ A + p 2 ν. f dµ = p A + p 2 f dµ = α and h(µ, σ) λ(µ, σ) = p 2h(ν, σ) p 2 λ(ν, σ) which completes the proof. = h(ν, σ) λ(ν, σ) For α (A, A 2 ) and ɛ > 0 we take ν E σ (Σ) such that h(ν,σ) λ(ν,σ) dim H Λ ɛ. By applying Lemma 7 we can find a measure µ M σ (Σ) such that f dµ = α and h(µ,σ) λ(µ,σ) dim H Λ 2ɛ. By combining Lemmas 2 and 3 we can find a measure η E σ (Σ) such that f dη = α and h(η,σ) λ(η,σ) dim H Λ 2ɛ. It thus follows that η(λ α ) = and by using Lemma 4 we can see that dim H Λ α dim H Λ 2ɛ. To complete the proof for the second case where α {A, A 2 we follow a similar approach to that used by Gelfert and Rams in [3].

15 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 5 Our strategy is to look at sequences with alternate between a hyperbolic measure of large dimension and the parabolic measure at the fixed point. We arrange that they spend more time at the fixed point and so this will determine the Birkhoff average. However if the proportion of time at the fixed point does not grow to quickly with relation to the proportion of time at the hyperbolic measure then the dimension can be given by the hyperbolic measure. Without loss of generality, we may assume that α = A, with the notation preceding Lemma 7. Then ω is a parabolic fixed points and f dδ = α and λ(δ, σ) = 0. We start by taking a measure µ E σ (Σ) such that f dµ = β for some β α which also satisfies λ(µ, σ) > 0. We now combine these two ergodic measures to find a new (non-invariant) measure with high dimension but for which the Birkhoff averages of f tend to α at almost every point. For ɛ > 0 and N, we define Ω(ɛ, N) to be the set of ω Σ such that for all n N (5.) (5.2) (5.3) A n f(ω) B(β, ɛ), A n g(ω) B(λ(µ, σ), ɛ), n log µ([ω,..., ω n ]) B(h(µ, σ), ɛ). It follows from the Birkhoff Ergodic Theorem, the Shannon McMillan Breiman Theorem and Egorov s Theorem that for any fixed δ > 0, we can find a decreasing sequence ɛ i > 0, such (5.4) µ(ω(ɛ i, i)) δ. where lim i ɛ i = 0 By the uniformity of the conclusion of Lemma and since var n A n f and var n A n g uniformly decrease to zero, we can, for i =, 2,..., choose another ɛ i decreasing to zero, so that for all ω Σ and all n i we have (5.5) (5.6) (5.7) var n A n f(ω) ɛ i var n A n g(ω) ɛ i λ n (ω) A n g(ω) + ɛ i. Finally, let ɛ i = max{ɛ i, ɛ i and let Ω(ɛ i) = Ω(ɛ i, i). We also need another sequence of integers {k i such that lim i k i = but (5.8) lim i k i ɛ i = 0.

16 6 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT For each i we define two measures ν i M(F i ) and η i M(F ik i ) where ν i simply gives equal weight to any cylinder containing an element of Ω(ɛ i ) and η i is simply the Dirac measure on the cylinder [,,,..., ] of length ik i. For q N let n q = q i= i( + k i). Define the probability η M(Σ) to be the distribution of a sequence of independent blocks that alternately have distribution ν i and η i, for i =, 2, 3,.... That is, let η = [ q= νq σ n q η q σ n q +q ]. This measure is not invariant however the proportion of the time it spends at the fixed point (,,,...) will converge to so the behaviour of the Birkhoff averages of a function f for a typical point will be governed by their value at (,,,...). Lemma 8.. For η almost all ω we have lim n A nf(ω) = α. 2. dim H η Π = h(µ,σ) λ(µ,σ) Proof. Note that since f is bounded and (5.9) lim q n q n q n q = 0 to show part we can just consider the subsequence n q. It follows from the definition of η that for η-almost all ω q i= i(β 2ɛ i) + q i= k ii(α 2ɛ i ) A nq f(ω) n q q i= i(β + 2ɛ i) + q i= k ii(α + 2ɛ i ). n q q i= The first part then follows since lim i q i=i = 0 and lim k ii(α + ɛ i ) = α. q n q q n q For the second part recall that for any measure ν if for ν-almost all x lim inf r 0 log νb(x, r) log r then dim H ν s. We let ν = η Π and consider s log η[ω,..., ω n ] and n λ n (ω). By using condition (5.3) and (5.8), we may deduce that the entropy of the distribution of the independent blocks (ω ni, ω ni +,..., ω ni +i ) is at least log( δ) +

17 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 7 i(h(µ, σ) ɛ i ). Since we use the uniform distribution, we may deduce that for all ω (5.0) log η([ω,..., ω nq ]) q log( δ) + q i(h(µ, σ) ɛ i ). Note that q log( δ) is dominated by q i= ih(µ, σ) for large q. For estimating the diameters of the cylinders, we note that i= (5.) n q λnq (ω) = n q j=0 i= g(σ j ω) + n q ɛ q (by (5.7)) q n i g(σ j ω) + ɛ i i( + k i ) (since ɛ i ɛ q ) j=n i q ( i Ai g(σ n i ω) + (ik i ) A iki g(σ ni +i ω) + ɛ i i( + k i ) ). i= Since the [σ n i ω] i contains an ω from Ω(ɛ i ) and since σ ni +i ω [ω 0 ] iki by (5.2) and (5.6), that (5.) is less than we obtain, (5.2) q (i(λ(µ, σ) + ɛ i + var i g) + ik i (A iki g(ω 0 ) + var iki g) + ɛ i i( + k i )) i= q i(λ(µ, σ) + 3ɛ i + 2ɛ i k i ). i= Recall that by condition (5.8) lim i ɛ i k i = 0. We now fix q > 0 and let ( ) ( ) q q+ exp i(λ(µ, σ) + 3ɛ i + 2ɛ i k i ) r > exp i(λ(µ, σ) + 3ɛ i + 2ɛ i k i ). i= Consider a ball B(x, r) where Πω = x. It follows from the definition of r that B(x, r) can intersect at most 3 cylinders of length n q which carry positive η-measure. Thus using the definition of r and (5.0) we have i= log ν(b(x, r)) log r log 3 + q log( δ) + q i= i(h(µ, σ) ɛ i) q i= i(λ(µ, σ) + 3ɛ i + 2ɛ i k i ) + (q + )(λ(µ, σ) + 3ɛ q+ + 2ɛ q+ k q+ ). Using condition (5.8) we can observe that this is of the form o(q 2 ) + q i= i(h(µ, σ) o()) o(q 2 ) + q. i= i(λ(µ, σ) + o())

18 8 A. JOHANSSON, T.M. JORDAN, A. ÖBERG AND M. POLLICOTT Since as q, r 0 this means for ν almost all x log ν(b(x, r)) h(µ, σ) lim r 0 log r λ(µ, σ) which completes the proof. Acknowledgment. The second author was supported by the EPSRC. The third author was supported by a STINT Fellowship at Amherst College during the Fall Semester of References [] L. Barreira, B. Saussol, Variational principles and mixed multifractal spectra, Trans. Amer. Math. Soc. 353 (200), no. 0, [2] W. Byrne, Multifractal Analysis of Parabolic Rational Maps, Phd Thesis, The University of North Texas. [3] K. Gelfert and M. Rams, Multifractal analysis of Lyapunov exponents of parabolic iterated function systems, preprint. [4] F. Hofbauer and P. Raith, The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull. 35 (992), no., [5] M. Kesseböhmer, B. Stratmann. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergodic theory and dynamical systems 24 (2004), no., [6] L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003), [7] Y. Pesin, Dimension Theory in Dynamical Systems, Contemporary Views and Applications, Chicago Lectures in Mathematics. University of Chicago Press, Chicago 997. [8] Y. Pesin and H. Weiss, The multifractal analysis of Birkhoff averages and large deviations, Global analysis of dynamical systems, 49 43, Inst. Phys., Bristol, 200. [9] K. Nakaishi, Multifractal formalism for some parabolic maps, Ergodic theory and dynamical systems, 20 (2000), [0] M. Urbański, Parabolic Cantor sets. Fund. Math. 5 (996), no. 3, [] P. Walters, An Introduction to Ergodic Theory, Springer, 982. [2] F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic theory and dynamical systems 23 (2003), no., Anders Johansson, Divsion of Mathematics and Statistics, University of Gävle, SE Gävle, Sweden address: ajj@hig.se Thomas M. Jordan, Department of Mathematics, University of Bristol, Bristol BS8 TW, United Kingdom address: thomas.jordan@bristol.ac.uk

19 MULTIFRACTAL ANALYSIS OF NON-UNIFORMLY HYPERBOLIC SYSTEMS 9 Anders Öberg, Department of Mathematics, Uppsala University, P.O. Box 480, SE Uppsala, Sweden address: anders@math.uu.se Mark Pollicott, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom address: mpollic@maths.warwick.ac.uk