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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 $\psi:Y\to X$ that induces an affine bijection $\psi^*$ 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., $\psi^*$ 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., Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2312.12562, We correct an error in the proof of Theorem 1. We have changed the title and the introduction, in order to emphasis the existence of Furstenberg-Weiss type almost 1-1 extensions that preserve 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 $\Sigma^G$, unlike $G=\mathbb{Z}$., Comment: There is an error in theorem 14, which we believe cannot be corrected. We are working on a new version of this topic, which may take a little longer

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 $(X,\sigma,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures $\nu_1, \cdots,\nu_r$ such that for every $1\leq i\leq r$ the measure-theoretic dynamical system $(X,\sigma,G,\nu_i)$ 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., Comment: 29 pages

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., Comment: 38pp; improvements to the Introduction made

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)., Comment: This is submitted to an upcoming book aimed at a general audience. Corrections after the referees comments are included

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., Comment: 13 pages

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.

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., Comment: Proofs simplified, references added

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., Comment: 18 p

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., Comment: 22 pages, 5 figures

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 $(X,T)$., Comment: This is a new version of the old paper "Set of invariant measures of generalized Toeplitz subshifts". In this version the results are more general

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., Comment: Minor corrections and a more detailed proof of Lemma 12 and of Lemma 13

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 $\sigma$-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 Sierpi\'nski gasket and carpet" tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $\sigma$-finite invariant measure, which is locally finite and unique up to scaling., Comment: 43 pages

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 $\omega$-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 $- \ln |f'|$., Comment: 34 pages. Minor changes. Accepted for publication in the Annales de l'Institut Henri Poincare, Analyse non lineaire

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., Comment: 15 pages

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 $-\ln |f'|$., Comment: 30 pages, minor corrections, to appear in Israel J. Math

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.

In this paper we prove that if two self-similar tiling systems, with respective stretching factors $\lambda_1$ and $\lambda_2$, have a common factor which is a non periodic tiling system, then $\lambda_1$ and $\lambda_2$ are multiplicatively dependent.

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., Comment: Added references, changed statement for Theorem 3.1