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Tomasz Downarowicz

Publications and source records attributed to Tomasz Downarowicz.

At least 19 recordsLinked to original sources
arXiv2026

Dynamical Cantor Staircase Functions and The Small Flow Boundary Property

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, \textit{flow-generated entropy}, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.

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arXiv2026

On a theorem Dan Rudolph: Part II: Amenable groups

We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli. We also address a relative version of this theorem.

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arXiv2025

On preservation of normality and determinism under arithmetic operations

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, which states that a number $y$ is deterministic if and only if $x+y$ is normal for every normal number $x$. Our approach is based on the notions of lower and upper entropy of a point in a topological dynamical system. The ergodic approach to Rauzy theorem naturally leads to the study of various aspects of normality and determinism in the general framework of dynamics of endomorphisms of compact metric groups. In particular, we generalize Rauzy theorem to ergodic toral endomorphisms. Also, we show that the phenomena described by Rauzy do not occur when one replaces the base $2$ normality associated with the $(\frac12,\frac12)$-Bernoulli measure by the variant of normality associated with a $(p,1-p)$-Bernoulli measure, where $p\neq\frac12$. Finally, we present some rather nontrivial examples which show that Rauzy-type results are not valid when addition is replaced by multiplication.

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arXiv2025

Universality of G-subshifts with specification

Let $G$ be an infinite countable amenable group and let $(X,G)$ be a $G$-subshift with specification, containing a free element. We prove that $(X,G)$ is universal, i.e., has positive topological entropy and for any free ergodic $G$-action on a standard probability space, $(Y,ν,G)$, with $h(ν)<h_{top}(X)$, there exists a shift-invariant measure $μ$ on $X$ such that the systems $(Y,ν,G)$ and $(X,μ,G)$ are isomorphic. In particular, any $K$-shift (consisting of the indicator functions of all maximal $K$-separated sets) containing a free element is universal.

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arXiv2023

Destruction of CPE-normality along deterministic sequences

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 $s_1<s_2<\dots$) "preserves (destroys) $μ$-normality" if, for any $x=(x_1,x_2,\dots)\inΛ^{\mathbb N}$ generic for $μ$, the sequence $x|_S=(x_{s_1},x_{s_2},\dots)$ is (is not) generic for $μ$. It is known from Kamae and Weiss that if $μ$ is i.i.d. then any deterministic set of positive lower density preserves $μ$-normality. We show that deterministic sets, except ones with a very primitive structure that we call "superficial", destroy $μ$-normality for any non-i.i.d. measure $μ$ with completely positive entropy (CPE). This generalizes Heersink and Vandehey's result for arithmetic progressions and the Gauss measure (associated to the continued fraction transformation). We give several examples showing that, outside the class of measures with CPE, $μ$-normality preservation can coexist with nearly any combination of three parameters: determinism of $S$ (or its lack), entropy of $μ$ (zero or positive), and disjointness (or its lack) between $μ$ and the measures derived from $S$.

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arXiv2023

Lifting generic points

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 $μ$ ergodic. Let $y\in Y$ be quasi-generic for $ν$. Then there exists a point $x\in X$ generic for $μ$ such that the pair $(x,y)$ is quasi-generic for $ξ$. This is a generalization of a similar theorem by T.\ Kamae, in which $(X,T)$ and $(Y,S)$ are full shifts on finite alphabets.

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arXiv2023

Multiorders in amenable group actions

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 $\tilde{\mathcal{O}}$ of linear orders of type $\mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Every free measure-preserving $G$-action $(X,μ,G)$ has a~multiorder $(\tilde{\mathcal{O}},ν,G)$ as a factor and has the same orbits as the $\mathbb Z$-action $(X,μ,S)$, where $S$ is the \emph{successor map} determined by the multiorder factor. Moreover, the sub-sigma-algebra $Σ_{\tilde{\mathcal{O}}}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $\mathbb Z$-action $(\tilde{\mathcal{O}},ν,\tilde S)$ a factor of $(X,μ,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $Σ_{\tilde{\mathcal{O}}}$, is preserved by the orbit equivalence with $(X,μ,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to $ h(μ,T,\mathcal P)=H(μ,\mathcal P|\mathcal{P}^-)$ known for $\mathbb Z$-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a~sub-sigma-algebra $Σ$, as soon as the ``orbit change'' is measurable with respect to $Σ$. In our variant, we replace the measurability assumption by a~simpler one: $Σ$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we provide a characterization of the Pinsker sigma-algebra of any $G$-process in terms of an appropriately defined remote past arising from a multiorder.

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arXiv2023

Asymptotic pairs in topological actions of amenable groups

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 topological $G$-action of positive entropy, then for every multiorder $(\tilde{\mathcal O},ν,G)$ and $ν$-almost every order $\prec\,\in\tilde{\mathcal O}$ there exists a $\prec$-asympotic pair in $X$. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of~$\mathbb Z$). We also prove that for every countable amenable group $G$, and every multiorder on $G$ arising from a tiling system, every topological $G$-action of entropy zero has an extension which has no $\prec$-asymptotic pairs for any $\prec$ belonging to this multiorder. Together, these two theorems give a characterization of topological $G$-actions of entropy zero: $(X,G)$ has topological entropy zero if and only if, for any multiorder $\tilde{\mathcal O}_{\boldsymbol{\mathsf T}}$ on $G$ arising from a tiling system of entropy zero, there exists an extension $(Y,G)$ of $(X,G)$, which has no $\prec$-asymptotic pairs for any $\prec\,\in\tilde{\mathcal O}_{\boldsymbol{\mathsf T}}$, equivalently, there exists a multiorder $(\tilde{\mathcal O},ν,G)$ on $G$, such that for $ν$-almost any $\prec\,\in\tilde{\mathcal O}$, there are no $\prec$-asymptotic pairs in $(Y,G)$.

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arXiv2021

Uniform continuity of entropy rate with respect to the $\bar f$-pseudometric

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 continuous when one endows the set of all frequency-typical sequences with the $\bar f$ pseudometric. As a consequence, we obtain the same result for the $\bar d$ pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.

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arXiv2021

Pure strictly uniform models of non-ergodic measure automorphisms

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 measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.

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arXiv2021

When all points are generic for ergodic measures

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 uniquely (alternatively, strictly) ergodic subsystems, - the map sending ergodic measures to their topological supports is continuous, - the Cesaro means of every continuous function converge uniformly.

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arXiv2020

The comparison property of amenable groups

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 $\mathsf A\preccurlyeq \mathsf B$), if there exists a finite partition $\mathsf A=\bigcup_{i=1}^k \mathsf A_i$ of $\mathsf A$ into clopen sets and there are elements $g_1,g_2,\dots,g_k$ in $G$ such that $g_1(\mathsf A_1), g_2(\mathsf A_2),\dots, g_k(\mathsf A_k)$ are disjoint subsets of $\mathsf B$. We say that the action \emph{admits comparison} if for any clopen sets $\mathsf A, \mathsf B$, the condition, that for every $G$-invariant probability measure $μ$ on $X$ we have the sharp inequality $μ(\mathsf A)<μ(\mathsf B)$, implies $\mathsf A\preccurlyeq \mathsf B$. Comparison has many desired consequences for the action, such as the existence of tilings with arbitrarily good Følner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $\mathbb z$-actions, extends to actions which admit comparison. We also study a purely group-theoretic notion of comparison: if every action of $G$ on any zero-dimensional compact metric space admits comparison then we say that $G$ has the \emph{comparison property}. Classical groups $\mathbb z$ and $\mathbb z^d$ enjoy the comparison property, but in the general case the problem remains open. In this paper we prove this property for groups whose every finitely generated subgroup has subexponential growth.

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arXiv2020

A fresh look at the notion of normality

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 $(F_n)$-normality coincides with the classical notion. We prove that: $\bullet$ If $(F_n)$ is a Følner sequence in $G$, such that for every $α\in(0,1)$ we have $\sum_n α^{|F_n|}<\infty$, then almost every $x\in\{0,1\}^G$ is $(F_n)$-normal. $\bullet$ For any Følner sequence $(F_n)$ in $G$, there exists an Cham\-per\-nowne-like $(F_n)$-normal set. $\bullet$ There is a natural class of "nice" Følner sequences in $(\mathbb N,\times)$. There exists a Champernowne-like set which is $(F_n)$-normal for every nice Følner \sq. $\bullet$ Let $A\subset\mathbb N$ be a classical normal set. Then, for any Følner sequence $(K_n)$ in $(\mathbb N,\times)$ there exists a set $E$ of $(K_n)$-density $1$, such that for any finite subset $\{n_1,n_2,\dots,n_k\}\subset E$, the intersection $A/{n_1}\cap A/{n_2}\cap\ldots\cap A/{n_k}$ has positive upper density in $(\mathbb N,+)$. As a consequence, $A$ contains arbitrarily long geometric progressions, and, more generally, arbitrarily long "geo-arithmetic" configurations of the form $\{a(b+ic)^j,0\le i,j\le k\}$. $\bullet$ For any Følner \sq\ $(F_n)$ in $(\mathbb N,+)$ there exist uncountably many $(F_n)$-normal Liouville numbers. $\bullet$ For any nice Følner sequence $(F_n)$ in $(\mathbb N,\times)$ there exist uncountably many $(F_n)$-normal Liouville numbers.

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arXiv2020

Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

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 sequence $(b_{n_i})_{i\in\mathbb N}$ is normal, if and only if $(n_i)_{i\in\mathbb N}$ is a deterministic sequence. Given a countable cancellative amenable semigroup $G$, and a Følner sequence $\mathcal F=(F_n)_{n\in\mathbb N}$ in $G$, we introduce the notions of normality preservation, determinism and subexponential complexity for subsets of $G$ with respect to $\mathcal F$, and show that for sets of positive lower $\mathcal F$-density these three notions are equivalent. The proof utilizes the apparatus of the theory of tilings of amenable groups and the notion of tile-entropy. We also prove that under a natural assumption on $\mathcal F$, positive lower $\mathcal F$-density follows from normality preservation. Finally, we provide numerous examples of normality preserving sets in various semigroups

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arXiv2020

Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures

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 occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $μ$ on $X$, whose restriction to the space of measures is compatible with the weak-$\star$ topology. Next, in this combined metric space we fix an open set $\mathcal U$ containing all ergodic measures, and we say that a block $B$ is "ergodic" if $B\in\mathcal U$. In this paper we prove the following main result: Given $\varepsilon>0$, every $x\in X$ decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set $M$ of coordinates of upper Banach density smaller than $\varepsilon$, all blocks in the decomposition are ergodic. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how $x\in X$ is partitioned into blocks (as long as their lengths are sufficiently large and bounded), after ignoring a set $M$ of upper Banach density smaller than $\varepsilon$, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set $M$, in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts $X\subsetΛ^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Følner sequence while the decomposition of $x\in X$ into blocks (of which majority is ergodic) is obtained with the help of a congruent system of tilings.

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arXiv2019

Symbolic extensions of amenable group actions and the comparison property

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 symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $\mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $\mathcal M_G(X)$ equals the extension entropy function $h^π$ of a symbolic extension $π:(Y,G)\to (X,G)$, where $h^π(μ)=\sup\{h_ν(Y,G): ν\inπ^{-1}(μ)\}$ ($μ\in\mathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $\mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET).

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