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We study the uniform ergodicity property for non-invertible topological and measure-preserving dynamical systems. It is shown that for topological dynamical systems uniform ergodicity is equivalent to eventually periodicity and that for…

Motivated by Sarnak's conjecture on M\"obius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation…

We define weaker forms of topological and measure theoretical equicontinuity for topological dynamical systems and we study their relationships with systems with discrete spectrum and zero sequence entropy. In the topological category we…

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are…

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). In order to do this we introduce mean equicontinuity…

This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological…

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel…

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of…

This paper studies ergodic properties of certain measures arising in the dynamics of holomorphic correspondences. These measures, in general, are not invariant in the classical sense of ergodic theory. We define a notion of ergodicity, and…

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study…

While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of…

We study the set of harmonic limits of empirical measures in topological dynamical systems. We obtain a characterization of unique ergodicity based of logarithmic (harmonic) mean convergence in place of Ces\`aro convergence. We introduce…

There is studied an invariant measure structure of a class of ergodicl discrete dynamical systems by means of the measure generating function method

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full…

We propose a new way of thinking about one parameter persistence. We believe topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between…

We introduce the notion of W-measurable sensitivity, which extends and strictly implies canonical measurable sensitivity, a measure- theoretic version of sensitive dependence on initial conditions. This notion also implies pairwise…

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new…

We study the dynamics of the metrics generated by measure preserving transformations. We consider a sequence of average metrics and define the corresponding sequence of $\epsilon$-entropies ({\it scaling sequence}) of the measure with…

We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into…

A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…