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Mateusz Więcek

Publications and source records attributed to Mateusz Więcek.

6 recordsLinked to original sources
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.

math.DS↗
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.

math.DS↗
arXiv2023

Existence of asymptotic pairs in positive entropy group actions

We provide a definition of a $\prec$-asymptotic pair in a topological action of a countable amenable group $G$, where $\prec$ is an order on $G$ of type $\mathbb Z$. We then prove that if $(\tilde{\mathcal O},ν,G)$ is a multiorder on $G$, then for every topological $G$-action of positive entropy there exists an $\prec$-asympotic pair for almost every order $\prec\,\in\tilde{\mathcal O}$. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of $\mathbb Z$).

math.DS↗
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 $(Y,G)$.

math.DS↗
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.

math.DS↗