Sebastián Donoso

University of Chile · Departamento de Ingeniería Matemática

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form \((T_{1}^{p_{1,j}(n)}\cdots T_{d}^{p_{d,j}(n)})_{n\in\mathbb{Z}...

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...

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...

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)^{\el...

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 reco...

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 gro...

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 reco...

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$ . We study the largeness of sets of the form $$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}...

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 t...

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T_1^{p_{1,j}(n)}\cdot\ldots\cdot T_d^{p_{d,j}(n)})_{n∈\ma...

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n...

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...

Let $(X,{\mathcal B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \big\{n\in\mathbb{N}\colon\mu(A\cap T^{-f_1(n)}A\cap T^{-f_2(n)}A\cap\ldots\cap T^{-f_k(n)}A)> \mu(A)^{k+1} - \epsilon \big\} \end{split} \end{equation*} for vario...

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ 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...

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 $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap...

On automorphism groups of Toeplitz subshifts, Discrete Analysis 2017:11, 19 pp. A discrete dynamical system is a space $X$ with some kind of structure, together with a map $\sigma\colon X\to X$ that preserves the structure. (For instance, if $X$ is a topological space, then one asks for $\sigma$ to be continuous, and if $X$ is a differentiable man...

We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ 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 $\mu$-a.e. $x\in X$ as $N\to\inf...

We show that if $(X,\mathcal{X},\mu,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\] converges for $\mu$-a.e. $x\in X$ as $N\to \infty$ for $f_0,f_1, f_2\in L^\infty(\mu)$. We also show that if $(X,...

In this article we show that any ergodic rigid system can be topologically realized by a uniformly rigid and (topologically) weak mixing topological dynamical system.

Cette thèse est consacrée à l'étude des différents problèmes liés aux structures des cubes , en théorie ergodique et en dynamique topologique. Elle est composée de six chapitres. La présentation générale nous permet de présenter certains résultats généraux en théorie ergodique et dynamique topologique. Ces résultats, qui sont associés d'une certain...

In this article we study the automorphism group Aut(X, σ) of a minimal subshift (X, σ) of low word complexity. In particular, we prove that Aut(X, σ) is virtually Z for aperiodic minimal subshifts with affine complexity on a subsequence, more precisely, the quotient of this group by the one gen-erated by the shift map is a finite group. In addition...

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\li...

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 G...

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 theore...

An $\infty$-step nilsystem is an inverse limit of minimal nilsystems. In this article is shown that a minimal distal system is an $\infty$-step nilsystem if and only if it has no nontrivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without nontrivial pairs with arbitrarily long finite IP...