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Maria Isabel Cortez

Publications and source records attributed to Maria Isabel Cortez.

8 recordsLinked to original sources
arXiv2019

On the centralizers of minimal aperiodic actions on the Cantor set

In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal $\mathbb Z$ action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group $G$, the centralizer of any minimal aperiodic $G$-action on the Cantor set is a subgroup of the centralizer of a minimal $\mathbb Z$-action.

math.DS↗
arXiv2014

Eigenvalues and strong orbit equivalence

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 dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated to (X,T) is trivial, the quotient group I(X,T)/E(X,T) is torsion free. We give examples with non trivial infinitesimal subgroups where this property fails. We also provide some realization results.

math.DS↗
arXiv2009

Choquet simplices as spaces of invariant probability measures of post-critical sets

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-critical set of a logistic map. Here, the post-critical set of a logistic map is the $ω$-limit set of its unique critical point. In fact we show the logistic map $f$ can be taken in such a way that its post-critical set is a Cantor set where $f$ is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential $- \ln |f'|$.

math.DS↗
arXiv2008

Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems

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 this question.

math.DS↗
arXiv2004

Rotation topological factors of minimal $\ZM^{d}$-actions on the cantor set

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 and we derive in this more general setting, a necessary and sufficient condition, which involves natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one these two types.

math.DS↗