MULTIFRACTAL ANALYSIS OF GIBBS MEASURES FOR NON-UNIFORMLY EXPANDING MAPS

We will consider the local dimension spectrum of a Gibbs measure on a non-uniformly hyperbolic system of MannevillePomeau type. We will present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. All three presentations are well known for uniformly hyperbolic systems. The theory of multifractal analysis for hyperbolic conformal dynamical systems is now extremely well developed. There are complete results for local dimension of Gibbs measures, Lyapunov spectra and Birkhoff spectra. For a general description see [14] and for more specific and very general results see [1] and [13]. In the situation of non-uniformly hyperbolic systems there have also been several papers however there is no such complete picture. In the case of local dimensions there are results by Nakaishi on the measure of maximal entropy, [12], some results by Pollicott and Weiss [15], by Kesseböhmer and Stratmann ([9] and [10]), by Yuri in [19] and in the case of complex dynamics by Bryne, [2]. The aim of this paper is to obtain a complete spectra for the local dimension of Gibbs measures for non-uniformly hyperbolic systems. We will be looking at cases where there exist parabolic periodic points but no critical points. Well known examples of such maps include the Manneville-Pomeau map and the Farey map. The methods used will be adapted from the papers [5] and [8] where the Lyapunov and Birkhoff spectra of such maps are considered. 1. Notation and results We consider non-uniformly expanding one-dimensional Markov maps. More precisely, let I = [0, 1]. Let {Ii}, i = 1, . , p be closed subintervals of I with disjoint interiors. Let A be a p× p matrix consisting of The research of M.R. was supported by grants EU FP6 ToK SPADE2, EU FP6 RTN CODY and MNiSW grant ’Chaos, fraktale i dynamika konforemna’. The reserach was started during a visit of M. R. to Bristol. M. R. would like to thank Bristol University for the hospitality shown during his visit. 1 2 THOMAS JORDAN AND MICHAL RAMS 0’s and 1’s where there exists k ∈ N such that for all i, j A(i, j) > 0. Let Ti : I → R be C bijective diffeomorphisms with closed domains Ji ⊂ Ii for which Ij ⊂ Ti(Ji) if A(i, j) = 1 and Ti(Ji) ∩ intIj = ∅ if A(i, j) = 0. We will let Λ0 = ∪iJi and let T : Λ0 → I be defined as Ti on Ji. When Ji ∩ Jj = {x} and i < j we will take T (x) = T (x). We will assume that |T ′ i (x)| ≥ 1 at every point x ∈ Ji and that there are at most countably many points with derivative ±1. We will denote by Λ the set of points whose trajectory never leaves Λ0. We will assume the existence of at least one periodic orbit {x, T (x), . , Tm−1(x), T(x) = x} where the derivative of T will be±1 at all points on the orbit. This means that the map is non-uniformly expanding. Let Σ be the full shiftspace with the usual left shift σ and ΣA ⊂ Σ the topologically transitive subshift with respect to the matrix A. Let Π : ΣA → Λ be defined by Π(i) = lim n→∞ T−1 i1 ◦ · · ·T −1 in (Λ) (the limit is always one point). The local dimension of a measure μ at a point x is defined by dμ(x) = lim r→0 log ν(B(x, r)) log r when this limit exists. For the rest of the paper let ν̃ be the Gibbs measure for a Hölder potential φ and ν = ν̃ ◦ Π−1, our goal will be to describe the local dimension spectrum function α→ dimHXα, where Xα = {x ∈ Λ : dν(x) = α} and α ∈ [0,∞]. ν is clearly invariant and ergodic under the map T . We will assume that P (φ) = 0 and φ(x) < 0 at each x ∈ ΣA. If this assumption is not satisfied then we can add a coboundary and a constant to φ to give a new function which does satisfy the assumption and has the same equilibrium state (see Theorem 9 in [3] and note that Gibbs measures for Hölder potentials on subshifts have strictly positive entropy). Let us introduce the following notation, let MT (Λ) be the set of T-invariant probability measures on Λ and Mσn(Σ) be the set of σ invariant probability measures on σ. For μ ∈ Mσ(ΣA) we will let h(μ, σ) denote the entropy of μ with respect to σ (recall by Abramov’s Theorem nh(μ, σ) = h(μ, σ) for μ ∈ Mσ(ΣA) ). Let ψ : ΣA → R be defined by ψ(i) = log |T ′ i1(Π(σi))| and for μ ∈ Mσ(ΣA) we define the average Lyapunov exponent to be