arXiv subjects

María Isabel Cortez

Publications and source records attributed to María Isabel Cortez.

17 recordsLinked to original sources

arXiv2026

Group actions with almost normal stabilizers

In this paper, we consider minimal group actions of countable groups on compact Hausdorff spaces by homeomorphisms. We show that the existence of a point with finite stabilizer imposes strong restrictions on the dynamics: the residual set of points then has stabilizers conjugate to the same almost normal subgroup of the acting group. Under certain conditions on the acting group such an action also has no essential holonomy. If the acting group is residually finite, we show that every group action with finite almost normal stabilizers with no essential holonomy arises as an almost finite-to-one factor of an essentially free action.

math.DS↗

arXiv2026

Induced dynamics and quasifactors for minimal equicontinuous actions on Stone spaces

A minimal equicontinuous action of a group $G$ on a Stone space $X$ is called a subodometer. If such a subodometer arises from a group rotation, we refer to it as an odometer. For subodometers $(X,G)$ we show that the hyperspace $\mathcal{H}(X)$ - given by all closed subsets of $X$ and the Vietoris topology - decomposes into subodometers. We show that an infinite subodometer is an odometer if and only if $\mathcal{H}(X)$ decomposes into factors of $(X,G)$. Similarly, we consider $\mathcal{M}(X)$, the space of regular Borel probability measures equipped with the weak-* topology. We show that for a subodometer $(X,G)$ also the connected space $\mathcal{M}(X)$ decomposes into subodometers. We prove that an infinite subodometer $(X,G)$ is an odometer if and only if $\mathcal{M}(X)$ decomposes into factors of $(X,G)$. For this, we study different notions of regular recurrence. Furthermore, we study the disjointness of minimal actions to subodometers and show that this disjointness can be detected from the pairwise disjointness of finite factors. Using this we prove that a minimal action is disjoint from all subodometers if and only if it has a connected maximal equicontinuous factor.

math.DS↗

arXiv2026

Minimal Equicontinuous Actions on Stone Spaces

In this article we study minimal equicontinuous actions on Stone spaces, which we call \emph{subodometers}, and do neither assume that the space is metrizable, nor any assumptions on the acting group. We show that the set of eigenvalues is a complete invariant for subodometers. Furthermore, we characterize minimal rotations on Stone spaces, which we call \emph{odometers}, via the intersection stability of their sets of eigenvalues. We show that any non-empty family of odometers allows for a minimal common extension and a maximal common factor, that both are odometers and that they are unique up to conjugacy. We provide examples that a similar statement does not hold for subodometers. We show that subodometers are given as inverse limits of minimal finite actions, that odometers are given as inverse limits of minimal finite rotations, and present how the minimal common extension and the maximal common factor of a non-empty family of odometers can be represented as an inverse limit. We establish that a minimal action $X$ is a subodometer if and only if its Ellis semigroup $E(X)$ is an odometer, and present how an inverse limit representation of $E(X)$ can be derived from the representation of $X$. Furthermore, we establish the existence of a universal odometer that has all subodometers as factors; as well as the existence of a maximal subodometer factor, and a maximal odometer factor of a given minimal action.

math.DS↗

arXiv2026

Stabilized automorphism groups and full groups of odometers

In this article, we show that the stabilized automorphism group of free exact odometers arising from actions of finitely generated residually finite groups coincides with the topological full group of the odometer acting on itself by right multiplication. We then prove that two free exact odometers have isomorphic stabilized automorphism groups if and only if they have isomorphic clopen subgroups of the same index. As a consequence, continuous orbit equivalence implies isomorphic stabilized automorphism groups, while for free $\mathbb{Z}^d$-odometers, isomorphic stabilized automorphism groups imply orbit equivalence. In general, neither continuous orbit equivalence nor orbit equivalence is equivalent to having isomorphic stabilized automorphism groups.

math.GR↗

arXiv2025

Constructing Toeplitz arrays via cut and project schemes

We show a one-to-one correspondence between Toeplitz arrays over residually finite topological groups and model sets obtained via specific cut and project schemes, built from the odometer associated to the Toeplitz array. As an application, we construct irregular Toeplitz arrays which are extensions of maximal rank $k$ over their maximal equicontinuous factor (given by the associated odometer) and have exactly $k$ different ergodic measures. A modification of the construction also allows to obtain examples with the same measure-theoretic structure, but infinite maximal rank.

math.DS↗

arXiv2025

A note on the structural stability of almost one-to-one maps

A continuous surjection $π:X\to Y$ between compact Hausdorff spaces induces continuous surjections $\mathcal{M}(π)\colon \mathcal{M}(X)\to\mathcal{M}(Y)$ and $\mathcal{H}(π): \mathcal{H}(X)\to\mathcal{H}(Y)$ between the spaces of regular Borel probability measures, and the spaces of closed subsets, respetively. It is well known that $\mathcal{H}(π)$ is irreducible if and only if $π$ is irreducible. We show that $\mathcal{M}(π)$ is irreducible if and only if $π$ is irreducible. Furthermore, we show that whenever $π$ is almost one-to-one then $\mathcal{M}(π)$ and $\mathcal{H}(π)$ are almost one-to-one. In particular, we observe that continuous surjections between compact metric spaces are almost one-to-one if and only if $\mathcal{H}(π)$ is almost one-to-one and a similar statement about $\mathcal{M}(π)$. Finally, we give alternative proofs for some results in 'Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps' by Xiongping Dai and Yuxuan Xie regarding semi-open maps.

math.DS↗

arXiv2024

Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability

Let $G$ be an infinite residually finite group. We show that for every minimal equicontinuous Cantor system $(Z,G)$ with a free orbit, and for every minimal extension $(Y,G)$ of $(Z,G)$, there exist a minimal almost 1-1 extension $(X,G)$ of $(Z,G)$ and a Borel equivariant map $ψ:Y\to X$ that induces an affine bijection $ψ^*$ between $M(Y,G)$ and $M(X,G)$, the spaces of invariant probability measures of $(Y,G)$ and $(X,G)$, respectively. If $Y$ is a Cantor set, then $(Y,G)$ and $(X,G)$ are Borel isomorphic, i.e., $ψ^*$ is also a homeomorphism. As an application, we show that the family of Toeplitz subshifts is a test for amenability for residually finite groups, i.e., a residually finite group $G$ is amenable if and only if every Toeplitz $G$-subshift has invariant probability measures.

math.DS↗

arXiv2023

Almost automorphic systems have invariant measures

Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this partially answers a question posed by Veech). In particular, every Toeplitz $G$-subshift has a non-empty space of invariant measures, meaning that this family of subshifts is not a test for amenability for countable groups. We prove that almost one-to-one extensions without measures ensure the existence of symbolic almost one-to-one extensions with equal characteristics. As a consequence, we obtain the most general result of this paper. Finally, as a corollary of our results, we deduce that the class of Toeplitz subshifts is not dense in the space of infinite transitive subshifts of $Σ^G$, unlike $G=\mathbb{Z}$.

math.DS↗

arXiv2023

Invariant measures of Toeplitz subshifts on non-amenable groups

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we construct a Toeplitz $G$-subshift $(X,σ,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $ν_1, \cdots,ν_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,σ,G,ν_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.

math.DS↗

arXiv2022

Settled elements in profinite groups

Given a polynomial of degree d over a number field, the image of the associated arboreal representation of the absolute Galois group of the field is a profinite group acting on the d-ary tree. Boston and Jones conjectured that for a quadratic polynomial, the image of such a representation contains a dense set of settled elements. Here an element is settled if it exhibits a certain pattern of growth of cycles at finite levels of the tree. In this paper, we prove the conjecture of Boston and Jones generically in the case when the quadratic polynomial has a strictly pre-periodic post-critical orbit of length 2, and provide new evidence that the conjecture holds for quadratic polynomials with strictly pre-periodic post-critical orbits of length at least 3. To prove our results, we introduce a new dynamical method, which uses the notions of a maximal torus and its Weyl group. These notions are analogous to the notions of maximal tori and Weyl groups in the theory of compact Lie groups, where they are fundamental. For profinite groups, maximal tori and Weyl groups contain the information about settled elements, and this is the foundation of our method.

math.DS↗

arXiv2021

Now that we are together: Biography of the Chilean Collective of Women Mathematicians

In this article we propose to give an account of the history of the Chilean collective of women mathematicians. We will begin by describing the context of the mathematical community in Chile and the process of forming the Collective, together with the first objectives we set ourselves. Then we will continue with an analysis of some reasons that support the need to create a group formed by mathematical women and also the fact of choosing horizontality and autonomy as structural pillars of our organization. On the other hand, we will refer to the main activities that we have carried out and we will provide an overview of the mathematical women's organizations that exist in Latin America. Finally, we will conclude by talking about the research project to which some of the members of the collective are dedicated nowadays (in particular the authors of this article).

math.HO↗

arXiv2017

Invariant measures for actions of congruent monotileable amenable groups

In this paper we show that for every congruent monotileable amenable group $G$ and for every metrizable Choquet simplex $K$, there exists a minimal $G$-subshift, which is free on a full measure set, whose set of invariant probability measures is affine homeomorphic to $K$. If the group is virtually abelian, the subshift is free. Congruent monotileable amenable groups are a generalization of amenable residually finite groups. In particular, we show that this class contains all the infinite countable virtually nilpotent groups. This article is a generalization to congruent monotileable amenable groups of one of the principal results shown in \cite{CP} for residually finite groups.

math.DS↗

arXiv2016

On Virtual Conjugacy of Generalized Odometers

The paper is focused on the study of continuous orbit equivalence for generalized odometers (profinite actions). We show that two generalized odometers are continuously orbit equivalent if and only if the acting groups have finite index subgroups (having the same index) whose actions are piecewise conjugate. This result extends M.~Boyle's flip-conjugacy theorem originally established for $\mathbb Z$-actions. As a corollary we obtain a dynamical classification of the restricted isomorphism between generalized Bunce-Deddens $C^*$-algebras. We also show that the full group associated with a generalized odometer is amenable if and only if the acting group is amenable.

math.DS↗

arXiv2015

Some examples of non-rectifiable, repetitive Delone sets

We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non bi-Lipschitz equivalent to the standard lattice). More generally, we construct such a set so that the translation action on the closure of its orbit has any prescribed Choquet simplex as its set of invariant probability measures (in particular, we provide uniquely ergodic examples). The construction relies on classical examples of Burago-Kleiner and McMullen, for which we give pure combinatorial (an effective) versions that have interest by themselves.

math.MG↗

arXiv2012

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

We show that for every metrizable Choquet simplex $K$ and for every group $G$, which is infinite, countable, amenable and residually finite, there exists a Toeplitz $G$-subshift whose set of shift-invariant probability measures is affine homeomorphic to $K$. Furthermore, we get that for every integer $d\geq 1$ and every Toeplitz flow $(X,T)$, there exists a Toeplitz ${\mathbb Z}^d$-subshift which is topologically orbit equivalent to $(X,T)$.

math.DS↗

arXiv2010

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $σ$-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the "integer Sierpiński gasket and carpet" tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $σ$-finite invariant measure, which is locally finite and unique up to scaling.

math.DS↗

arXiv2008

Invariant measures of minimal post-critical sets of logistic maps

We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map $f$ can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of $f$ for the potential $-\ln |f'|$.

math.DS↗