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Tomasz Downarowicz

Publications and source records attributed to Tomasz Downarowicz.

At least 37 records · Page 2Linked to original sources
arXiv2019

A strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function

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 average along every infinite arithmetic progression (the Möbius function is an example of such a \sq\ $y$) then for every $N\ge 2$ there exists a subshift $Σ$ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is \emph{uniform}, i.e., for any continuous function $f:Σ\to {\mathbb R}$ and every $ε>0$ there exists $n_0$ such that for any $n\ge n_0$ and any $x\inΣ$ we have $$ \left|\frac1n\sum_{i=1}^{n}f(T^ix)\,y_i\right|<ε. $$ More importantly, by a fine-tuned modification of the construction from (DS) we create a \emph{strictly ergodic} subshift, with all the desired properties of the example in (DS) (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $\log N$). The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemanczyk in the context of the so-called strong MOMO (Möbius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.

math.DS↗
arXiv2019

Empirical approach to the x2, x3 conjecture

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 finding atomic measures whose supports are far from being uniformly distributed, and we used computer software to discover a number of such measures (which we call \emph{outlier measures}). The structure of these measures indicates the possibility that a sequence of atomic measures may converge to a non-Lebesgue measure; likely one which is a combination of the Lebesgue measure and one or more atomic measures.

math.DS↗
arXiv2018

Zero-dimensional isomorphic dynamical models

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 of extreme points $\mathsf{ex}K$ is a countable union $\bigcup_n E_n$, where each set $E_n$ is compact, zero-dimensional, and the restriction of $Φ$ to the Bauer simplex $K_n$ spanned by $E_n$ can be `embedded' in some topological dynamical system, then $Φ$ can be `realized' in a zero-dimensional system.

math.DS↗
arXiv2017

Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits

For a topological dynamical system $(X, T)$ we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal cardinality of uniform generators has lead us to new challenges in the theory of symbolic extensions. At the beginning we show that uniform generators can be identified with so-called symbolic extensions with an embedding, i.e., symbolic extensions admitting an equivariant measurable selector from preimages. From here we focus on such extensions and we strive to characterize the collection of the corresponding extension entropy functions on invariant measures. For aperiodic zero-dimensional systems we show that this collection coincides with that of extension entropy functions in arbitrary symbolic extensions, which, by the general theory of symbolic extensions, is known to coincide with the collection of all affine superenvelopes of the entropy structure of the system. In particular, we recover, after [Bu16], that an aperiodic zero-dimensional system is asymptotically h- expansive if and only if it admits an isomorphic symbolic extension. Next we pass to systems with periodic points, and we introduce the notion of a period tail structure, which captures the local growth rate of periodic orbits. Finally, we succeed in precisely identifying the wanted collection of extension entropy functions in symbolic extensions with an embedding: these are all the affine superenvelopes of the usual entropy structure which lie above certain threshold function determined by the period tail structure. This characterization allows us, among other things, to give estimates (and in examples to compute precisely) the optimal cardinality of a uniform generator.

math.DS↗
arXiv2017

Dynamical quasitilings of amenable group

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 properties, i.e to every $x\in X$ we can assign a quasitiling $\mathcal{T}_x$ of $G$ (with all the $\mathcal{T}_x$ using the same, finite set of shapes) such that the tiles of $\mathcal{T}_x$ are disjoint, their union has arbitrarily high lower Banach Density, all the shapes of $\mathcal{T}_x$ are large subsets of an arbitrarily large Følner set, and if we consider $\mathcal{T}_x$ to be an element of a shift space over a certain finite alphabet, then the mapping $x\mapsto \mathcal{T}_x$ is a factor map.

math.DS↗
arXiv2016

Dynamics in dimension zero. A survey

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 switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli--Vershik systems and give for it a sufficient condition.

math.DS↗
arXiv2016

Almost full entropy subshifts uncorrelated to the Möbius function

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 close to $\log N$, uncorrelated to $y$. In particular, for $y=μ$ being the Möbius function, we get that there exist subshifts as above which satisfy the assertion of Sarnak's conjecture. The existence of positive entropy systems uncorrelated to the Möbius function is claimed in Sarnak's survey \cite{sarnak} (and attributed to Bourgain), however, to our knowledge no examples have ever been published. We fill in this gap and by the way we show that this has nothing to do with more advanced algebraic properties (for instance multiplicativity) of the considered sequence.

math.DS↗
arXiv2015

Minimal spaces with cyclic group of homeomorphisms

There are two main subjects in this paper. 1) For a topological dynamical system $(X,T)$ we study the topological entropy of its "functional envelopes" (the action of $T$ by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of $X$). In particular we prove that for zero-dimensional spaces $X$ both entropies are infinite except when $T$ is equicontinuous (then both equal zero). 2) We call $Slovak$ $space$ any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems $(X,T)$ with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.

math.DS↗
arXiv2015

Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles

The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shift-invariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining $ξ$ of two shift-invariant measures $μ$ and $ν$, every point $x$ generic for $μ$ lifts to a pair $(x,y)$ generic for $ξ$ (such $y$ exists in the full symbolic space). This lemma allows us to translate correlation between bounded sequences to the language of correlation of measures. Finally, to establish that the property of an invariant measure being uncorrelated to any ergodic measure is essentially weaker than the property of being disjoint from any ergodic measure, we develop and apply criteria for ergodicity of four-jump cocycles over irrational rotations. We believe that apart from the applications to studying the notion of correlation, the two developed tools: the lifting lemma and the criteria for ergodicity of four-jump cocycles, are of independent interest. This is why we announce them also in the title. In the Appendix we also introduce the notion of conditional disjointness.

math.DS↗
arXiv2015

Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy

We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the "infimum fomula" gives correct value of topological entropy.

math.DS↗
arXiv2015

Isomorphic extensions and applications

If $π:(X,T)\to(Z,S)$ is a topological factor map between uniquely ergodic topological dynamical systems, then $(X,T)$ is called an isomorphic extension of $(Z,S)$ if $π$ is also a measure-theoretic isomorphism. We consider the case when the systems are minimal and we pay special attention to equicontinuous $(Z,S)$. We first establish a characterization of this type of isomorphic extensions in terms of mean equicontinuity, and then show that an isomorphic extension need not be almost one-to-one, answering questions of Li, Tu and Ye.

math.DS↗
arXiv2015

Tilings of amenable groups

We prove that for any infinite countable amenable group $G$, any $ε> 0$ and any finite subset $K\subset G$, there exists a tiling (partition of $G$ into finite "tiles" using only finitely many "shapes"), where all the tiles are $(K; ε)$-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of $G$ (in the sense that the mappings, associated to different from unity elements of $G$, have no fixpoints), on a zero-dimensional space, and which has topological entropy zero.

math.GR↗
arXiv2015

Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture

Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension. In particular, we reestablish (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do all generalized Sturmian subshifts (not only the classical Sturmian subshift). We also give an example of an irregular Toeplitz subshift which fits our criterion. We give an example of a model of an odometer which is not even Toeplitz (it is weakly mixing), hence does not fit our criterion. However, for this example, we manage to produce a separate proof of Sarnak's conjecture. Next, we provide a class of Toeplitz sequences which fail Sarnak's conjecture (in a weak sense); all these examples have positive entropy. Finally, we examine the example of a Toeplitz sequence from [AKL] (which fails Sarnak's conjecture in the strong sense) and prove that it has positive entropy, as well (this proof has been announced in [AKL]). This paper can be considered a sequel to [AKL], it also fills some gaps of [D].

math.DS↗
arXiv2012

Measure-theoretic chaos (Chaos au sens de la mesure)

We define new isomorphism-invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaoses {\rm DC2} and its slightly stronger version (which we denote by {\rm DC}{\small$1\tfrac12$}). We prove that: 1. If a \tl\ system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is \tl ly {\rm DC2} ({\rm DC}{\small$1 \tfrac12$}) chaotic. 2. Every ergodic system with positive Kolmogorov--Sinai entropy is measure-theoretically$^+$ chaotic (even in a bit stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, a system of entropy zero with uniform measure-theoretic$^+$ chaos. \bigskip \centerline{\bf Résumé} Nous introduisons de nouveaux invariants pour les systèmes dynamiques définis sur des espaces probabilisés standards, appelés respectivement {\rm chaos mesuré} et {\rm chaos$^+$ mesuré}. Ces notions sont des analogues du chaos topologique {\rm DC2} et de l'une de ses variantes, renforcée, que nous appelons {\rm DC}{\small$1 \tfrac12$}. Nous montrons d'une part que si un système dynamique topologique est chaotique (resp. chaotique$^+$) au sens da la mesure relativement à\ l'une de ses mesures invariantes ergodiques, alors il l'est du point de vue topologique au sens correspondant. Nous montrons que tout système ergodique d'entropie métrique positive est chaotique$^+$ au sens de la mesure (même en un sens plus fort, i.e. {\rm uniformément}). Nous donnons enfin un exemple de système dynamique topologique d'entropie nulle qui présente pour l'une de ses mesures invariantes ergodiques un chaos$^+$ mesuré uniforme.

math.DS↗
arXiv2012

Rank as a function of measure

We establish certain topological properties of rank understood as a function on the set of invariant measures on a topological dynamical system. To be exact, we show that rank is of Young class LU (i.e., it is the limit of an increasing sequence of upper semicontinuous functions)

math.DS↗
arXiv2008

The law of series

We consider an ergodic process on finitely many states, with positive entropy. Our first main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. That is, the occurrences of so understood "rare event" $B$ along the time axis can appear either with gap sizes of nearly exponential distribution (like in the independent Bernoulli process), or they "attract" each-other. Our second main result states that a {\it typical} ergodic process of positive entropy has the following property: the distribution functions of the normalized hitting times for the majority of cylinders $B$ of lengths $n'$ converge to zero along a \sq\ $n'$ whose upper density is 1. The occurrences of such a cylinder $B$ "strongly attract", i.e., they appear in "series" of many frequent repetitions separated by huge gaps of nearly complete absence. These results, when properly and carefully interpreted, shed some new light, in purely statistical terms, independently from physics, on a century old (and so far rather avoided by serious science) common-sense phenomenon known as {\it the law of series}, asserting that rare events in reality, once occurred, have a mysterious tendency for untimely repetitions.

math.PR↗
arXiv2008

Spontaneous clustering in theoretical and some empirical stationary processes

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical "unbiased behavior" with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for "rare events" defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

math.PR↗