We study recurrent operators from a new perspective by introducing the notion of hyper-recurrent operators and establish robust connections with quasi-rigid operators. For example, we prove that a recurrent operator on a separable Banach space is quasi-rigid if and only if it is a linear factor of a hyper-recurrent operator, and show that the quasi-rigid operators found in Costakis, Manoussos and Parissis's work, along with many others, are, in fact, hyper-recurrent operators. Furthermore, we provide a negative answer, using a class of operators introduced by Tapia, to the question by Costakis et al. whether $T \oplus T$ is recurrent whenever $T$ is.

We discuss relations between the amenability of a graph and spectral properties of a random walk driven by a dynamical system. In order to include graphs which are not locally compact, we introduce the concept of amenability of weighted graphs, which generalises the usual notion as the new definition is shown to be equivalent to Foelner's condition. As a first result, we obtain the following generalisation of Kesten's amenability criterion to graphs and non-independent increments: If the random walk is driven by a full-branched Gibbs-Markov map, the graph is amenable with respect to the weight induced by the random walk if and only if the spectral radius of the associated Markov operator is equal to one. By employing inducing schemes, one then obtains criteria for amenability through Markov maps with less regularity. We conclude the paper with the following applications to Schreier graphs. If the random walk is driven by an uniformly expanding map with non-Markovian increments, then, under certain conditions, the Schreier graph is amenable if the probability of a return in time n does not decay exponentially in n. Furthermore, in the context of geometrically finite Kleinian groups, one obtains a version of Brooks's amenability criterion for not necessarily normal subgroups.

We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or Rényi entropy). We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalise the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined zeta-function at its critical parameter. We then generalise Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the zeta-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

We prove uniform Ancona—Gouëzel—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 Hölder equivalent. As application, we identify the set of minimal δ-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. In this setting, the uniformity allows to identify the visual boundary with minimal, δ-conformal densities in the sense of Patterson with respect to the exponent of convergence δ. This is of interest as δ coincides with the Hausdorff dimension of the radial limit set. Moreover, this extends Roblin’s identification of s-conformal densities for s > δ.

AbstractWe find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys.186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys.116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose nth partial quotient is bigger than α n , where (α n ) is a sequence such that ∑1/α 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.

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 Carathéodory-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.

We show the existence of invariant ergodic sigma-additive probability measures with full support on X for a class of linear operators L : X -> X, where L is a weighted shift operator and X either is the Banach space c(0)(R) l(p)(R) 1 <= p < infinity. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Holder continuous potential A:X -> R

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is a general Borel standard space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as 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 do not have a purely additive part. We then apply our results to the construction of invariant measures of time-homogeneous 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.

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of \(\sigma \)-finite conformal measures and give applications to the construction of harmonic functions.

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 : {-1, 1}(N) -> R of the form f(x(1), x(2),...) = x(1) + 2(-gamma) x(2) + 3(-gamma) x(3) + ... + n(-gamma) x(n) + ... with gamma > 1. For 3/2 < gamma <= 2, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist almost surely (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

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.

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.

We correct a flaw in the proof of Proposition 6.3 in [1].

We employ techniques from optimal transport in order to prove the decay of transfer operators associated with 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 -> R a given fixed Holder function, and denote by L the Ruelle operator associated with 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.

We give a corrected statement and extended proof of Theorem 4.1 in [2] and an alternative proof of Corollary 4.1 which is independent of the wrong statement in the theorem.

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 Holder continuous and the topological Markov chain has big images and preimages. Finally, an application to periodic hyperbolic manifolds is given. (C) 2012 Elsevier Inc. All rights reserved.