Michał Rams

Rich phase transitions in step skew products

We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated with the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behaviour of different types. However, in some sense the expanding behaviour is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent.

The Dimension of Projections of Fractal Percolations

Fractal percolation or Mandelbrot percolation is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by

Multifractal analysis of weak Gibbs measures for non-uniformly expanding C1 maps

E\subset\mathbb{R}^{2}

Multifractal analysis for Bedford-McMullen carpets

a typical realization of the fractal percolation on the plane, If

Packing dimension estimation for exceptional parameters

\dim_{\rm H}E<1

Lyapunov spectrum for multimodal maps

then for all lines ℓ the orthogonal projection Eℓ of E to ℓ has the same Hausdorff dimension as E,If

Increasing digit subsystems of infinite iterated function systems

\dim_{\rm H}E>1

Growth rates for geometric complexities and counting functions in polygonal billiards

then for any smooth real valued function f which is strictly increasing in both coordinates, the image f(E) contains an interval. The second statement is quite interesting considering the fact that E is almost surely a Cantor set (a random dust) for a large part of the parameter domain, see Chayes et al. (Probab. Theory Relat. Fields, 77(3):307–324, 1988). Finally, we solve a related problem about the existence of an interval in the algebraic sum of d≥2 one-dimensional fractal percolations.

Absolute continuity of the SBR measure for non-linear fat baker maps

We will consider the local dimension spectrum of a weak Gibbs measure on a C non-uniformly hyperbolic system of Manneville-Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth 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 early results on the local dimension spectra see [3] and [18], for a general description see [16] and for more specific and very general results see [1] and [15]. However, the picture is not complete for non-uniformly hyperbolic systems. There are results for specific nonuniformly expanding systems by Kessebohmer and Stratmann ([11] and [12]) and in the case of complex dynamics by Byrne, [2]. As our interest is in the local dimension spectrum for invariant measures, five papers are of special interest for us. The first results about local dimension were in the papers by Nakaishi and by Pollicott and Weiss, [14],[17]. They obtained the local dimension spectrum for the measure of maximal entropy for a restricted class of parabolic maps without critical points. Stratmann and Urbanski in [19] and [20] considered complex (hence analytic) maps with possible critical points. However, the analyticity assumption in these papers could be probably reduced to assumptions from section 3 of [7]. In Theorem 4 of [24] Yuri computes a portion of the multifractal spectra for weak Gibbs measures of some non-uniformly hyperbolic systems. 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 research 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 MICHA L RAMS The aim of this paper is to obtain a complete spectrum for the local dimension of weak Gibbs measures for C non-uniformly hyperbolic systems. We will include results about the points where the local dimension is infinite, a phenomenon which does not occur in the uniformly hyperbolic setting. We will be considering systems with parabolic periodic points but no critical points. Well known examples of such maps include the Manneville-Pomeau map and the Farey map. The methods we use are adapted from the papers [6] and [9] where the Lyapunov and Birkhoff spectra of such maps are considered. For most of the paper we work directly with the original system, without inducing, which lets us omit the usual assumptions about behaviour of the map around parabolic points (except for Theorem 4 which deals with analyticity of the spectra). 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 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) = Ti(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 allow the existence of parabolic periodic orbits {x, T (x), . . . , Tm−1(x), T(x) = x} where the derivative of T will be ±1 at all points on the orbit. The existence of a parabolic orbit implies that the map is non-uniformly expanding. Let Σ = {1, . . . , p}N be the full shiftspace with the usual left shift σ and ΣA ⊂ Σ the subshift with respect to the matrix A. For i ∈ Σ we will denote by in its n-th element and by i n the sequence of its first n elements. For some 0 < β < 1 we will use the metric dκ on ΣA given by dκ(i, j) = κ |i∧j| where |i ∧ j| = inf{m ∈ N : im 6= jm} − 1. By our assumptions, (ΣA, σ) is topologically transitive. Let Π : ΣA → Λ be defined by Π(i) = lim n→∞ T−1 i1 ◦ · · · ◦ T −1 in (Λ) LOCAL DIMENSION FOR NON-UNIFORMLY EXPANDING MAPS 3 (the limit is always one point, see the proof of Lemma 2.1 in [21]). The local dimension of a measure μ at a point x is defined by dμ(x) = lim r→0 log μ(B(x, r)) log r when this limit exists. Let φ : ΣA → R be a continuous potential. We call a σ-invariant probabilistic measure ν supported on ΣA weak Gibbs for the potential φ if there exists a constant P and a decreasing sequence {kn} such that limn→∞ kn = 0 and for every i ∈ ΣA and for every n exp(−nkn) ≤ ν([i1 . . . in]) exp(Snφ(i)− nP ) ≤ exp(nkn). This is similar to the definition in [23] except we require the result to hold for all sequences not just for ν-almost all sequences. The existence of such weak Gibbs measures for all continuous potentials is established in [10]. For the rest of the paper let ν be the weak Gibbs measure for the potential φ and ν = ν ◦ Π−1. Our goal will be to describe the local dimension spectrum function α→ dimHXα, where Xα = {x ∈ Λ : dν(x) = α}. Since ν is invariant under the map σ it follows that ν is invariant under T . We will assume that P (φ) = 0 and φ(x) < 0 at each x ∈ ΣA. Note here that this assumption is not very restrictive in the narrower class of Holder potentials. If φ is Holder continuous and does not satisfy this assumption then it is possible to add a constant and a coboundary to obtain a new potential which will satisfy these conditions (see Theorem 9 in [4] and note that Gibbs measures for Holder potentials on subshifts of finite type have positive entropy). Let us introduce the following notation, let MT (Λ) be the set of T-invariant probability measures on Λ and Mσn(ΣA) be the set of σ invariant probability measures on ΣA. For μ ∈ Mσ(ΣA) let h(μ, σ) denote the entropy of μ with respect to σ (recall that 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(i))| and for μ ∈ Mσ(ΣA) we define the Lyapunov exponent of the measure μ as

Almost complete Lyapunov spectrum in step skew-products

In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford–McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.