We consider a Ruelle-Perron-Frobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures. Key words: random dynamical system, random subshift of finite type, transfer operator, Gibbs measures, equilibrium states. 1991 Mathematics Subject Classification. Primary 58F03, 58F15; Secondary 60J10, 54H20. Introduction Random dynamical systems (RDS) generalize dynamical systems by allowing the dependence on an additional parameter evolving in time and describing a stochastic influence (cf. Arnold and Crauel [3]). The latter is modelled by an abstract dynamical system(\Ome...

In a series of three papers, we study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinite-dimensional analogues of Axiom A systems. Our main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. In this paper we develop a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. In the second, we treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems. The final paper contains the proof of the main theorem which asserts the existence and uniqueness of a natural spatio-temporal measure for certain weakly coupled circle map lattices with a natural coupling. Key words: spatially extended system, coupled map lattice, Axiom A, hyperbolicity, structural stability, stable manifold, shadowing lemma, Markov partition, Gibb...

Abstract. We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.

We show that one dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive some power of f is mixing and in particular the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, to the average rate at which typical points start to exhibit exponential growth of the derivative.

1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3

In [9] a classification theory for topological Markov chains (shifts of finite type, intrinsic Markov chains) appears which, although incomplete [10], provides algebraic criteria for deciding when two such chains are topologically conjugate. Numerous invariants effectively exclude topological conjugacy for many interesting examples.

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: .

Abstract. If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shift-invariant probability measures on V each of which corresponds to an α-invariant probability measure on X. Moreover, every α-invariant probability measure on X arises essentially in this way. The last part of the paper deals with the connection between the twosided beta-shift Vβ arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α arising from the companion matrix of the minimal polynomial of β, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures on Vβ and certain α-invariant probability measures on X. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms. 1.

We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state pathspace measure for which `the density' is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti-Cohen (local) fluctuation theorem and it is illustrated via a number of examples.

These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical ” nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fifty-year period under discussion and on numerous occasions was fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre. These notes are meant as “easy reading ” for a mathematically sophisticated reader familiar with most of the concepts which appear in the text. I recommend the book [59] as a source of background reading and the survey [44] (both written jointly with Boris Hasselblatt) as a reference source for virtually all necessary definitions and (hopefully) illuminating discussions. The origin of these notes lies in my talk at the dynamical systems branch of the huge conference held in Moscow in June of 2003 on the occasion of Kolmogorov’s