Tomasz Downarowicz
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View article: On preservation of normality and determinism under arithmetic operations
On preservation of normality and determinism under arithmetic operations Open
In this paper we develop a general ergodic approach which reveals the underpinnings of the effect of arithmetic operations involving normal and deterministic numbers. This allows us to recast in new light and amplify the result of Rauzy, w…
View article: Lifting generic points
Lifting generic points Open
Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y ,…
View article: Multiorders in amenable group actions
Multiorders in amenable group actions Open
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure \nu on the collection \widetilde{\mat…
View article: Lifting generic points
Lifting generic points Open
Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $ξ$ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $μ$ on $X$ and $ν$ on $Y$, with …
View article: Asymptotic pairs in topological actions of amenable groups
Asymptotic pairs in topological actions of amenable groups Open
We provide a definition of a $\prec$-asymptotic pair in a topological action of a countable group $G$, where $\prec$ is an order on $G$ of type $\mathbb Z$. We then prove that if $G$ is a countable amenable group and $(X,G)$ is a topologic…
View article: Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures
Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures Open
Consider a subshift over a finite alphabet, X\subset \Lambda^\mathbb{Z} (or X\subset\Lambda^{\mathbb{N}_0} ). With each finite block B\in\Lambda^k appearing in X we associate the empirical measure ascribing to every block C\in\Lambda^l the…
View article: Tail variational principle and asymptotic $h$-expansiveness for amenable group actions
Tail variational principle and asymptotic $h$-expansiveness for amenable group actions Open
In this paper we prove the tail variational principle for actions of countable amenable groups. This allows us to extend some characterizations of asymptotic $h$-expansiveness from $\mathbb{Z}$-actions to actions of countable amenable grou…
View article: Destruction of CPE-normality along deterministic sequences
Destruction of CPE-normality along deterministic sequences Open
Let $μ$ be a shift-invariant measure on $Λ^{\mathbb N}$, where $Λ$ is a finite or countable alphabet. We say that an infinite subset $S=\{s_1,s_2,\dots\}\subset\mathbb N$ (where $s_1
View article: Pure strictly uniform models of non-ergodic measure automorphisms
Pure strictly uniform models of non-ergodic measure automorphisms Open
The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any me…
View article: Uniform Continuity of Entropy Rate With Respect to the F-Pseudometric
Uniform Continuity of Entropy Rate With Respect to the F-Pseudometric Open
Assume that a sequence is frequency-typical for a finite-valued stationary stochastic process . We prove that the function associating to the entropy-rate of is uniformly continuous when one endows the set of all frequency-typical seq…
View article: Multiorders in amenable group actions
Multiorders in amenable group actions Open
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a~{\em multiorder} on a~countable group we mean any probability measure $ν$ on the collection $\tilde{\m…
View article: The Symbolic Extension Theory in Topological Dynamics
The Symbolic Extension Theory in Topological Dynamics Open
In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.
View article: Uniform continuity of entropy rate with respect to the $\bar f$-pseudometric
Uniform continuity of entropy rate with respect to the $\bar f$-pseudometric Open
Assume that a sequence $x=x_0x_1\ldots$ is frequency-typical for a finite-valued stationary stochastic process $\textbf X$. We prove that the function associating to $x$ the entropy-rate $\bar H(\textbf X)$ of $\textbf X$ is uniformly cont…
View article: Uniform continuity of entropy on the regular points endowed with $f$-bar
Uniform continuity of entropy on the regular points endowed with $f$-bar Open
Given a sequence $x=x_0x_1\ldots$ which is frequency-typical for a finite-valued stationary stochastic process, write $h(x)$ for the entropy of that process. We prove that the function $h$ is uniformly continuous when one endows the set of…
View article: Uniform continuity of entropy rate with respect to the $\\bar\n f$-pseudometric
Uniform continuity of entropy rate with respect to the $\\bar\n f$-pseudometric Open
Assume that a sequence $x=x_0x_1\\ldots$ is frequency-typical for a\nfinite-valued stationary stochastic process $\\textbf X$. We prove that the\nfunction associating to $x$ the entropy-rate $\\bar H(\\textbf X)$ of $\\textbf X$\nis unifor…
View article: A fresh look at the notion of normality
A fresh look at the notion of normality Open
Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) Følner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the $(F_n…
View article: Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures
Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures Open
Consider a subshift over a finite alphabet, $X\subset Λ^{\mathbb Z}$ (or $X\subsetΛ^{\mathbb N_0}$). With each finite block $B\inΛ^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\inΛ^l$ the frequency of o…
View article: Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation
Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation Open
A classical Kamae-Weiss theorem states that an increasing sequence $(n_i)_{i\in\mathbb N}$ of positive lower density is \emph{normality preserving}, i.e. has the property that for any normal binary sequence $(b_n)_{n\in\mathbb N}$, the seq…
View article: When all points are generic for ergodic measures
When all points are generic for ergodic measures Open
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 …
View article: A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function
A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function Open
A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnak's conjecture. More precisely, it is proved that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero aver…
View article: Symbolic extensions of amenable group actions and the comparison property
Symbolic extensions of amenable group actions and the comparison property Open
Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of s…
View article: Empirical approach to the x2, x3 conjecture
Empirical approach to the x2, x3 conjecture Open
We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\mod 1$ and by $3\mod 1$, since such measures play an important role in deciding Furstenberg's $\times 2, \times 3$ conjecture. Our specific focus was f…
View article: Zero-dimensional isomorphic dynamical models
Zero-dimensional isomorphic dynamical models Open
By an \emph{assignment} we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems, obeying some natural restrictions. We prove that if $Φ$ is an aperiodic assignment on a Choquet simplex $K$ such that the set o…
View article: Dynamics in dimension zero A survey
Dynamics in dimension zero A survey Open
The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can …
View article: The comparison property of amenable groups
The comparison property of amenable groups Open
Let a countable amenable group $G$ act on a \zd\ compact metric space $X$. For two clopen subsets $\mathsf A$ and $\mathsf B$ of $X$ we say that $\mathsf A$ is \emph{subequivalent} to $\mathsf B$ (we write $\mathsf A\preccurlyeq \mathsf B$…
View article: Almost Full Entropy Subshifts Uncorrelated to the Möbius Function
Almost Full Entropy Subshifts Uncorrelated to the Möbius Function Open
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 $Σ$ over $N$ symbols, with entropy arbitrarily c…
View article: Uniform generators, symbolic extensions with an embedding, and structure\n of periodic orbits
Uniform generators, symbolic extensions with an embedding, and structure\n of periodic orbits Open
For a topological dynamical system $(X, T)$ we define a uniform generator as\na finite measurable partition such that the symmetric cylinder sets in the\ngenerated process shrink in diameter uniformly to zero. The problem of\nexistence and…
View article: Dynamical quasitilings of amenable group
Dynamical quasitilings of amenable group Open
We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, Følner and dynamical prop…