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 Gurevič-pressures of the extension and the base coincide whereas the converse holds true if the potential is HÃűlder continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given.

Abstract. 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.Dezember 1, 2014.

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space X ≡ E N , where E is a general standard Borel space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on X and obtain the existence of equilibrium states as finitely additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states does not have a purely finite additive part.We then apply our results to the construction of invariant measures of timehomogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.2010 Mathematics Subject Classification. 37D35, 28Dxx.

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

We study a class of potentials f on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if f does not satisfy Bowen's condition.We apply these results to potentials f :with γ > 1. For 3/2 < γ ≤ 2, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.