Coupling methods for random topological Markov chains

Abstract

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.

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

“…It has already been noticed that the 1-Wasserstein distance issued from optimal transportation theory is very convenient to prove exponential contraction properties for Markov chains (see e.g. [HM08,Sta13,Oll09]). In this article, we observe that this idea applies very effectively to the dynamics of expanding maps: indeed the dual transfer operator of an expanding map with respect to a normalized potential can be seen as a Markov chain, for which we prove exponential contraction.…”

Exponential law for random subshifts of finite type

“…As in the proof of [?, Lemma 7.2], the assumption on the integrability of log κ implies that the above limit is finite a.s., say n−1 k=0 r k κ(θ n−k ω) ≤ c ω . However, it is also pointed out in [32] that if κ is integrable, then we have an a.s. uniform upper bound, say C φ on n−1 k=0 r k κ(θ n−k ω). Given a Hölder function ψ, then we define…”

On the Lyapunov spectrum of relative transfer operators

“…The article is structured as follows. In Section 2, we give the details with respect to the topology of H, introduce the Vaserstein metric and prove Theorem A by adaption of a result for non-stationary shift spaces in [10]. In Section 3, we consider stochastic cocycles and prove that dominated splittings for the induced action can be lifted.…”

“…The proof of the estimate is well-known and therefore omitted (see, e.g. equation (3) in [10]). However, note that the estimate has various important consequences and implies, e.g., that L m…”