arXiv subjects

Sebastián Donoso

Publications and source records attributed to Sebastián Donoso.

33 records · Page 2Linked to original sources

arXiv2022

Additive averages of multiplicative correlation sequences and applications

We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.

math.DS↗

arXiv2022

Directional mean dimension and continuum-wise expansive $\mathbb{Z}^k$-actions

We study directional mean dimension of $\mathbb{Z}^k$-actions (where $k$ is a positive integer). On the one hand, we show that there is a $\mathbb{Z}^2$-action whose directional mean dimension (considered as a $[0,+\infty]$-valued function on the torus) is not continuous. On the other hand, we prove that if a $\mathbb{Z}^k$-action is continuum-wise expansive, then the values of its $(k-1)$-dimensional directional mean dimension are bounded. This is a generalization (with a view towards Meyerovitch and Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite-dimensional.

math.DS↗

arXiv2022

Strong Orbit Equivalence and Superlinear Complexity

We show that within any strong orbit equivalent class, there exist minimal subshifts with arbitrarily low superlinear complexity. This is done by proving that for any simple dimension group with unit $(G,G^+,u)$ and any sequence of positive numbers $(p_n)_{n\in\mathbb{N}}$ such that $\lim n/p_n=0$, there exist a minimal subshift whose dimension group is order isomorphic to $(G,G^+,u)$ and whose complexity function grows slower than $p_n$. As a consequence, we get that any Choquet simplex can be realized as the set of invariant measures of a minimal Toeplitz subshift whose complexity grows slower than $p_n$.

math.DS↗

arXiv2021

Decomposition of multicorrelation sequences and joint ergodicity

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $\mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $\mathbb{Z}^{d}$-systems.

math.DS↗

arXiv2020

Interplay between finite topological rank minimal Cantor systems, $\mathcal S$-adic subshifts and their complexity

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable ${\mathcal S}$-adic subshifts. This is done establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like all classical zero entropy examples) have finite topological rank. Conversely, we analyze the complexity of ${\mathcal S}$-adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so called left to right ${\mathcal S}$-adic subshifts. We also exhibit that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank 2 subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.

math.DS↗

arXiv2019

Optimal lower bounds for multiple recurrence

Let $(X, \mathcal{B},μ,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $ε>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colonμ(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> μ(A)^{k+1} - ε\right\} \end{split} \end{equation*} for various families $\{f_1,\dots,f_k\}$ of sequences $f_i\colon \mathbb{N} \to \mathbb{N}$. For $k \leq 3$ and $f_{i}(n)=i f(n)$, we show that $S$ has positive density if $f(n)=q(p_n)$ where $q \in \mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1) =0$ and $p_n$ denotes the $n$-th prime; or when $f$ is a certain Hardy field sequence. If $T^q$ is ergodic for some $q \in \mathbb{N}$, then for all $r \in \mathbb{Z}$, $S$ is syndetic if $f(n) = qn + r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $k\geq 4$, and for $k = 3$ we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.

math.DS↗

arXiv2019

Directional dynamical cubes for minimal $\mathbb{Z}^{d}$-systems

We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$. We study the structural properties of systems that satisfy the so called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a $\mathbb{Z}^d$-system $(X,T_1,\ldots,T_d)$ that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal $\mathbb{Z}^d$-systems that enjoy the unique closing parallelepiped property and provide explicit examples.

math.DS↗

arXiv2017

Pointwise multiple averages for sublinear functions

For any measure preserving system $(X,\mathcal{B},μ,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fejér functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are in general bad for convergence on arbitrary systems, but they are good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

math.DS↗

arXiv2017

On automorphism groups of Toeplitz subshifts

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity bounded by $C n^{1+ε}$ whose automorphism groups are not finitely generated. Finally, we observe the coalescence and the automorphism group give no restriction on the complexity since we provide a family of coalescent Toeplitz subshifts with positive entropy such that their automorphism groups are arbitrary finitely generated infinite abelian groups with cyclic torsion subgroup (eventually restricted to powers of the shift).

math.DS↗

arXiv2017

A pointwise cubic average for two commuting transformations

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove that for a system $(X,μ,S,T)$ with commuting transformations $S$ and $T$, the average \[\frac{1}{N^2} \sum_{i,j=0}^{N-1} f_0(S^i x)f_1(T^j x)f_2(S^i T^j x)\] converges a.e. as $N$ goes to infinity for any $f_1,f_2,f_3\in L^{\infty}(μ)$.

math.DS↗

arXiv2017

Quantitative multiple recurrence for two and three transformations

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A\cap T_{3}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0.$$

math.DS↗

arXiv2016

Pointwise convergence of some multiple ergodic averages

We show that for every ergodic system $(X,μ,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n \prod_{j=1}^d T_j^{n_j}x)\cdots f_d(T_d^n \prod_{j=1}^d T_j^{n_j}x). \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. If $X$ is distal, we prove that the average \[\frac{1}{N}\sum_{i=0}^{N} f_1(T_1^nx)f_2(T_2^nx)\cdots f_d(T_d^nx) \] converges for $μ$-a.e. $x\in X$ as $N\to\infty$. We also establish the pointwise convergence of averages along cubical configurations arising from a system commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye.

math.DS↗

arXiv2015

On automorphism groups of low complexity subshifts

In this article we study the automorphism group ${\rm Aut}(X,σ)$ of subshifts $(X,σ)$ of low word complexity. In particular, we prove that Aut$(X,σ)$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a $d$-step nilsystem is nilpotent of order $d$ and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually $\mathbb{Z}$.

math.DS↗

arXiv2014

Dynamical cubes and a criteria for systems having products extensions

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have a less intricate dynamic and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions, and a computation of the automorphism group of the minimal Robinson tiling.

math.DS↗

arXiv2013

Enveloping semigroups of systems of order d

In this paper we study the Ellis semigroup of a d-step nilsystem and the inverse limit of such systems. By using the machinery of cubes developed by Host, Kra and Maass, we prove that such a system has a d-step topologically nilpotent enveloping semigroup. In the case d=2, we prove that these notions are equivalent, extending a previous result by Glasner.

math.DS↗