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Aurelia Dymek

Publications and source records attributed to Aurelia Dymek.

7 recordsLinked to original sources
arXiv2026

On automorphisms of $\mathscr{B}$-admissible and related subshifts

We adapt ideas of Kim and Roush [15], originally developed in the study of automorphisms of sofic subshifts, to obtain sufficient conditions under which a subshift has a huge automorphism group. We apply this approach to non-sofic subshifts defined by sets of multiples. In particular, we establish a dichotomy for the $\mathscr{B}$-admissible subshift: its automorphism group is either trivial or contains an embedded copy of the automorphism group of the full shift $\{0,1\}^{\mathbb Z}$. In the latter case, we say that the automorphism group is huge. We further show that the automorphism group of the hereditary closure of the $\mathscr{B}$-free subshift is huge whenever $\mathscr{B}\subset \mathbb N$ is infinite and contains no infinite pairwise coprime subset.

math.DS↗
arXiv2026

$\mathfrak{B}$-free integers in number fields and dynamics

In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the Möbius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\mathfrak{B}$-free integers in number fields, suggested 5 years later by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a number field $K$, let $\mathfrak{B}$ be a family of pairwise coprime ideals in its ring of integers $\mathcal{O}_K$, such that $\sum_{\mathfrak{b}\in\mathfrak{B}}1/|\mathcal{O}_K / \mathfrak{b}|<\infty$. We study the dynamical system determined by the set $\mathcal{F}_\mathfrak{B}=\mathcal{O}_K\setminus \bigcup_{\mathfrak{b}\in\mathfrak{B}}\mathfrak{b}$ of $\mathfrak{B}$-free integers in $\mathcal{O}_K$. We show that the characteristic function $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$ of $\mathcal{F}_\mathfrak{B}$ is generic along the natural Følner sequence for a probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a description of ``patterns'' appearing in $\mathcal{F}_\mathfrak{B}$ and compute the topological entropy of the orbit closure of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system has a non-trivial topological joining with an ergodic rotation on a compact Abelian group.

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arXiv2025

A note on $\mathscr{B}$-free sets and the existence of natural density

Given $\mathscr{B}\subseteq \mathbb{N}$, let $\mathcal{M}_\mathscr{B}=\bigcup_{b\in\mathscr{B}}b\mathbb{Z}$ be the correspoding set of multiples. We say that $\mathscr{B}$ is taut if the logarithmic density of $\mathcal{M}_\mathscr{B}$ decreases after removing any element from $\mathscr{B}$. We say that $\mathscr{B}$ is minimal if it is primitive (i.e.\ $b| b'$ for $b,b'\in\mathscr{B}$ implies $b=b'$) and the characteristic function $η$ of $\mathcal{M}_\mathscr{B}$ is a Toeplitz sequence (i.e.\ for every $n\in \mathbb{N}$ there exists $s_n$ such that $η$ is constant along $n+s_n\mathbb{Z}$). With every $\mathscr{B}$ one associates the corresponding taut set $\mathscr{B}'$ (determined uniquely among all taut sets by the condition that the associated Mirsky measures agree) and the minimal set $\mathscr{B}^*$ (determined uniquely among all minimal sets by the condition that every configuration appearing on $\mathcal{M}_{\mathscr{B}^*}$ appears on $\mathcal{M}_\mathscr{B}$: for every $n\in \mathbb{N}$, there exists $k\in \mathbb{Z}$ such that $\mathcal{M}_{\mathscr{B}^*}\cap [0,n]=\mathcal{M}_\mathscr{B} \cap[k,k+n]-k$). Besicovitch [2] gave an example of $\mathscr{B}$ whose set of multiples does not have the natural density. It was proved in [7, Lemma 4.18] that if $\mathcal{M}_{\mathscr{B}'}$ posses the natural density then so does $\mathcal{M}_\mathscr{B}$. In this paper we show that this is the only obstruction: every configuration $ijk\in \{0,1\}^3$ (with $ij\neq 01$), encoding the information on the existence of the natural density for the triple $\mathcal{M}_\mathscr{B},\mathcal{M}_{\mathscr{B'}},\mathcal{M}_{\mathscr{B}^*}$, can occur. Furthermore, we show that $\mathcal{M}_\mathscr{B}$ and $\mathcal{M}_{\mathscr{B}'}$ can differ along a set of positive upper density.

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arXiv2024

Invariant measures for $\mathscr{B}$-free systems revisited

For $ \mathscr{B} \subseteq \mathbb{N} $, the $ \mathscr{B} $-free subshift $ X_η $ is the orbit closure of the characteristic function of the set of $ \mathscr{B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_η $, have their analogues for $ X_η $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures ([Keller, G. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21, 3 (2021), Paper No. 2140008]). A central assumption in our work is that $ η^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_η $) is regular. From this we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_η $ from above and below.

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arXiv2022

Automorphisms of $\mathcal{B}$-free and other Toeplitz shifts

We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of $\mathcal{B}$-free subshifts satisfying them, extending [10]. On the other hand we provide an example of a $\mathcal{B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 in [13].

math.DS↗
arXiv2022

Minimality of $\mathfrak{B}$-free systems in number fields

Let $K$ be a finite extension of $\mathbb{Q}$ and $\mathcal{O}_K$ be its ring of integers. Let $\mathfrak{B}$ be a primitive collection of ideals in $\mathcal{O}_K$. We show that any $\mathfrak{B}$-free system is essentially minimal. Moreoever, the $\mathfrak{B}$-free system is minimal if and only if the characteristic function of $\mathfrak{B}$-free numbers is a Toeplitz sequence. Equivalently, there are no ideal $\mathfrak{d}$ and no infinite pairwise coprime collection of ideals $\mathcal{C}$ such that $\mathfrak{d}\mathcal{C}\subseteq\mathfrak{B}$. Moreover, we find a periodic structure in the Toeplitz case. Last but not least, we describe the restrictions on the cosets of ideals contained in unions of ideals.

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