An extension of Kesten's criterion for amenability to topological Markov chains

The main results of this note extend a theorem of Kesten for symmetric random walks on discrete groups to group extensions of topological Markov chains. In contrast to the result in probability theory, there is a notable asymmetry in the assumptions on the base. That is, it turns out that, under very mild assumptions on the continuity and symmetry of the associated potential, amenability of the group implies that the Gurevic-pressures of the extension and the base coincide whereas the converse holds true if the potential is H\"older continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given.Comment: New proof of Lemma 5.3 due to the gap in the first version of the articl

Coupling methods for random topological Markov chains

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages-property have a spectral gap.Comment: 17 page

On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system.Comment: Improved bounds for $(\alpha_n)

The Martin boundary of an extension by a hyperbolic group

We prove uniform Ancona-Gou\"ezel-Lalley inequalities for an extension by a hyperbolic group $G$ of a Markov map which allows to deduce that the visual boundary of the group and the Martin boundary are H\"older equivalent. As application, we identify the set of minimal conformal measures of a regular cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the covering group, provided that this group is hyperbolic

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact

On the Lyapunov spectrum of relative transfer operators

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the $C^0$-topology by positive matrices with an associated dominated splitting.Comment: The article now contains a section on decay of correlations of relative transfer operator

Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps

We employ techniques from optimal transport in order to prove decay of transfer operators associated to iterated functions systems and expanding maps, giving rise to a new proof without requiring a Doeblin-Fortet (or Lasota-Yorke) inequality. Our main result is the following. Suppose $T$ is an expanding transformation acting on a compact metric space $M$ and $A: M \to \mathbb{R}$ a given fixed H{\"o}lder function, and denote by $L$ the Ruelle operator associated to $A$. We show that if $L$ is normalized (i.e. if $L(1)=1$), then the dual transfer operator $L^*$ is an exponential contraction on the set of probability measures on $M$ with the $1$-Wasserstein metric.Our approach is flexible and extends to a relatively general setting, which we name Iterated Contraction Systems. We also derive from our main result several dynamical consequences; for example we show that Gibbs measures depends in a Lipschitz-continuous way on variations of the potential

Thermodynamic formalism for random non-uniformly expanding maps

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $C^1$-potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carath\'eodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.Comment: 58 pages, revised versio

On random topological Markov chains with big images and preimages

We introduce a relative notion of the 'big images and preimages'-property for random topological Markov chains. This then implies that a relative version of the Ruelle-Perron-Frobenius theorem holds with respect to summable and locally Hoelder continuous potentials.Comment: Corrected and extended version of the article published in Stochastics and Dynamics 201

Quenched and annealed equilibrium states for random Ruelle expanding maps and applications

In this paper we describe the spectral properties of semigroups of expanding maps acting on Polish spaces, considering both sequences of transfer operators along infinite compositions of dynamics and integrated transfer operators. We prove that there exists a limiting behaviour for such transfer operators, and that these semigroup actions admit equilibrium states with exponential decay of correlations and several limit theorems. The reformulation of these results in terms of quenched and annealed equilibrium states extend results by Baladi (1997) and Carvalho, Rodrigues & Varandas (2017), where the randomness is driven by a random walk and the phase space is assumed to be compact. Furthermore, we prove that the quenched equilibrium measures vary H\"older continuously and that the annealed equilibrium states can be recovered from the latter. Finally, we give some applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and on the boundary of equilibria