During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_w^n:{\lambda }_p\left(A\right)\to {\lambda }_p\left(A\right)$ defined on the Köthe sequence space ${\lambda }_p\left(A\right)$ exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$ and any $n\in \mathbb{N}$ is obtained. Under this assumption, the principal measure of $B_w^n$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $ϵ$-chaos for any $0<ϵ<\mathrm{diam}{\lambda }_p\left(A\right)$.

Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor /\left(\beta -1\right)]$. We show that $K_{\beta }$ has a unique mixing measure ${\nu }_{\beta }$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ${\nu }_{\beta }$ the digits ${\left(d_i\right)}_{i\ge 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta$-expansions....

Soit $(X,𝔛)$ un espace mesurable muni d’une transformation bijective bi-mesurable $\tau$. Soit $\varphi$ une application mesurable de $X$ dans un groupe localement compact à base dénombrable $G$. Nous notons ${\tau }_{\varphi }$ l’extension de $\tau$, induite par $\varphi$, au produit $X\times G$. Nous donnons une description des mesures positives ${\tau }_{\varphi }$-invariantes et ergodiques. Nous obtenons aussi une généralisation du théorème de réduction cohomologique de O.Sarig [5] à un groupe LCD quelconque.

Every aperiodic endomorphism $f$ of a nonatomic Lebesgue space which possesses a finite 1-sided generator has a 1-sided generator $\beta$ such that $k_f\le card\beta \le k_f+1$. This is the best estimate for the minimal cardinality of a 1-sided generator. The above result is the generalization of the analogous one for ergodic case.

We prove that on a metrizable, compact, zero-dimensional space every ${\mathbb{Z}}^d$-action with no periodic points is measurably isomorphic to a minimal ${\mathbb{Z}}^d$-action with the same, i.e. affinely homeomorphic, simplex of measures.

An effective construction of positive-entropy almost one-to-one topological extensions of the Chacón flow is given. These extensions have the property of almost minimal power joinings. For each possible value of entropy there are uncountably many pairwise non-conjugate such extensions.

We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1–22] and Rosiński and Żak [Stoc. Proc. Appl. 61 (1996) 277–288] from the univariate to the d-dimensional case. Thereafter, we show that multivariate Lévy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein − Uhlenbeck (supOU) processes or (fractionally integrated)...