Maria Isabel Cortez

Pontificia Universidad Católica de Chile | UC · Facultad de Matemática

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...

An automorphism of a rooted spherically homogeneous tree is settled if it satisfies certain conditions on the growth of cycles at finite levels of the tree. In this paper, we consider a conjecture by Boston and Jones that the image of an arboreal representation of the absolute Galois group of a number field in the automorphism group of a tree has a...

The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory. For instance, Giordano, Putnam and Skau have shown in [29] that the dimension group (see [24] for an introduction about dimension groups) of a minimal Z-action on the Cantor set compl...

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 sup...

In this article we study the centralizer of a minimal aperiodic action of a finitely generated group on the Cantor set (an aperiodic minimal Cantor system). We show that this centralizer is always an extension of some LEF groups, having as a consequence that the Thompson group $T$ can not be a subgroup of the centralizer of any aperiodic minimal Ca...

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 monotilea...

Cet article propose un panorama de la féminisation de la recherche en mathématiques au Chili, ainsi qu’un programme de recherche qui permette l’étude sociologique de ce champ scientifique. La première partie passe en revue les théories féministes de l’histoire des sciences ainsi que l’état de l’art en sociologie des mathématiques. La seconde partie...

The paper is focused on the study of continuous orbit equivalence for minimal equicontinuous systems. We establish that every equicontinuous system is topologically conjugate to a profinite action, where the finite-index subgroups are not necessarily normal. We then show that two profinite actions $(X,G)$ and $(Y,H)$ are continuously orbit equivale...

This article provides an overview of the feminization of research in mathematics in Chile and a research program that advances the sociological study of this scientific field. The first part reviews the feminist theories about the history of science and the state of the art in sociology of mathematics. The second part gives a first glimpse on the p...

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-conj...

Linearly repetitive Delone sets are the simplest aperiodic repetitive Delone sets of the Euclidean space, e.g. any self similar Delone set is linearly repetitive.We present here some combinatorial, ergodic and mixing properties of their associated dynamical systems. We also give a characterization of such sets via the patch frequencies. Finally, we...

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues E(X,T) of the minimal Cantor system (X,T) is a subgroup of the intersection I(X,T) of all the images of the dimens...

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 translatio...

We show that for every metrizable Choquet simplex K and for every group G, which is in�finite, countable, amenable and residually �finite, there exists a Toeplitz G-subshift whose set of shift-invariant probability measures is affi�ne homeomorphic to K. Furthermore, we get that for every integer d > 1 and every Toeplitz flow (X, T), there exists a...

We show that for every metrizable Choquet simplex $K$ and for every group $G$, which is amenable, finitely generated 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 minimal Cantor system $(X,T)$...

We show that every uniquely ergodic minimal Cantor system is topological orbit equivalent to the natural extension of a numeration scale associated to a logistic map.

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 inva...

We prove linearly repetitive Delone systems have finitely many Delone system factors up to conjugacy. This result is also applicable to linearly repetitive tiling systems.

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-cri...

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 i...

In this paper we recall the concepts of G-odometers and G-subodometers for G-actions, where G is a discrete finitely generated group; these generalize the notion of an odometer in the case G = ℤ. We characterize the G-regularly recurrent systems as the minimal almost one-to-one extensions of subodometers, from which we deduce that the family of the...

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts and odometers measure--theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to...

In this paper we prove that if two self-similar tiling systems, with respective stretching factors λ 1 and λ 2, have a common factor which is a nonperiodic tiling system, then λ 1 and λ 2 are multiplicatively dependent.

In this paper we show that, for every Choquet simplex K and for every d > 1, there exists a Z(d)-Toeplitz system whose set of invariant probability measures is affine homeomorphic to K. Then, we conclude that K may be realized as the set of invariant probability measures of a tiling system (Omega(T), R-d).

In this paper we give a definition of Toeplitz sequences and odometers for ℤd actions for d ≥ 1 which generalizes that in dimension one. For these new concepts we study properties of the induced Toeplitz dynamical systems and odometers classical for d = 1. In particular, we characterize the ℤd-regularly recurrent systems as the minimal almost 1-1 e...

In this paper we recall the concepts of $G$-odometer and $G$-subodometer for $G$-actions, where $G$ is a discrete finitely generated group, which generalize the notion of odometer in the case $G=\ZZ$. We characterize the $G$-regularly recurrent systems as the minimal almost 1-1 extensions of subodometers, from which we deduce that the family of the...

In this paper we study conditions under which a free minimal $\mz^d$-action on the Cantor set is a topological extension of the action of $d$ rotations, either on the product $\mt^d$ of $d$ 1-tori or on a single 1-torus $\mt^1$. We extend the notion of {\it linearly recurrent} systems defined for $\mz$-actions on the Cantor set to $\mz^d$-actions a...

2. Definitions and background 2.1. Dynamical systems. By a topological dynamical system we mean a couple (X,T) where X is a compact metric space and T : X → X is a homeomorphism. We say that it is a Cantor system if X is a Cantor space; that is, X has a countable basis of its topology which consists of closed and open sets (clopen sets) and does no...