We describe the homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological entropy and that they are actually semi-conjugate to an algebraic homeomorphism in the case when the entropy is positive.

For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and $[T,T^{-1}]$ automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...

Let $\beta >1$ be a non-integer. We consider 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 existence and uniqueness of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_p\otimes \lambda$, where $m_p$ is the Bernoulli measure on ${\{0,1\}}^{\mathbb{N}}$ with parameter $p$ ($0<p<1$) and $\lambda$ is the normalized Lebesgue measure on $[0,\lfloor \beta \rfloor (\beta -1\left)\right]$. Furthermore, this measure is of the form $m_p\otimes {\mu }_{\beta ,p}$, where ${\mu }_{\beta ,p}$ is equivalent to $\lambda$. We prove that the measure of maximal entropy and $m_p\otimes \lambda$ are mutually singular. In...

For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.

We develop a relative isomorphism theory for random Bernoulli shifts by showing that any random Bernoulli shifts are relatively isomorphic if and only if they have the same fibre entropy. This allows the identification of random Bernoulli shifts with standard Bernoulli shifts.

The paper deals with the notion of entropy for doubly stochastic operators. It is shown that the entropy defined by Maličky and Riečan in [MR] is equal to the operator entropy proposed in [DF]. Moreover, some continuity properties of the [MR] entropy are established.

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....