Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers

A class of strictly ergodic Toeplitz flows with positive entropies and trivial topological centralizers is presented

Constructions of cocycles over irrational rotations

We construct a coboundary cocycle which is of bounded variation, is homotopic to the identity and is Hölder continuous with an arbitrary Hölder exponent smaller than 1

On the asymptotic relation of topological amenable group actions

For a topological action $\Phi$ of a countable amenable orderable group $G$ on a compact metric space we introduce a concept of the asymptotic relation \A (\Phi) and we show that \A (\Phi) is non-trivial if the topological entropy $h(\Phi)$ is positive. It is also proved that if the Pinsker $\sigma$-algebra $\pi_{\mu}(\Phi)$ is trivial, where $\mu$ is an invariant measure with full support, then \A (\Phi) is dense. These results are generalizations of those of Blanchard, Host and Ruette (\cite{BHR}) that concern the asymptotic relation for $\Z$-actions. We give an example of an expansive $G$-action ($G=\Z^2$) with \A (\Phi) trivial which shows that the Bryant--Walters classical result (\cite{BW}) fails to be true in general case

An application of the ergodic theorem of information theory to Lyapunov exponents of cellular automata

We prove a generalization of the individual ergodic theorem of the information theory and we apply it to give a new proof of the Shereshevsky inequality connecting the metric entropy and Lyapunov exponents of dynamical systems generated by cellular automata