Related papers: A strictly ergodic, positive entropy subshift unif…

We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy…

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of…

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic…

It is proved that whenever a zero entropy dynamical system $(X,T)$ has only countably many ergodic measures and $\mu$ stands for the arithmetic M{\"o}bius function, then there exists a subset $A$ of integers depending only on the system, of…

We show that there are an irrational rotation $Tx=x+\alpha$ on the circle $\mathbb{T}$ and a continuous $\varphi\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting…

We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this…

Sarnak's M\"{o}bius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) \mu (n)= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing…

The M\"obius disjointness conjecture of Sarnak states that the M\"obius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the…

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. The Dyck and Motzkin shifts are well-known examples of transitive subshifts over a finite alphabet that are not…

We show that the M\"obius disjointess of zero entropy dynamical systems implies the existence of an increasing sequence of positive integers along which the Chowla conjecture on autocorrelations of the M\"obius function holds.

We prove that for every ergodic invariant measure with positive entropy of a continuous map on a compact metric space there is $\delta>0$ such that the dynamical $\delta$-balls have measure zero. We use this property to prove, for instance,…

We show that the $\mathscr{B}$-free subshift $(S,X_{\mathscr{B}})$ associated to a $\mathscr{B}$-free system is intrinsically ergodic, i.e.\ it has exactly one measure of maximal entropy. Moreover, we study invariant measures for such…

We construct a family of shift spaces with almost specification and multiple measures of maximal entropy. This answers a question from Climenhaga and Thompson [Israel J. Math. 192 (2012), no. 2, 785--817]. Elaborating on our examples we…

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of…

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient conditions are…

We present a streamlined proof of a result essentially present in previous work of the author, namely that for every set $S = \{s_1, s_2, \ldots\} \subset \mathbb{N}$ of zero Banach density and finite set $A$, there exists a minimal…

We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if…

Let f be a self-map of a compact manifold M, admitting an global SRB measure \mu. For a continuous test function \phi on M and a constant \alpha>0, consider the set of the initial points for which the Birkhoff time averages of the function…