arXiv subjects

Mauricio Achigar

Publications and source records attributed to Mauricio Achigar.

11 recordsLinked to original sources

arXiv2026

Algebraic expansivity on abelian groups

Building on the author's earlier work on topological and abstract expansivity, this paper introduces and explores the notion of algebraic expansivity for endomorphisms of abelian groups. We analyze the fundamental properties of this algebraic analogue, establish its relationship with Weiss's algebraic entropy, and prove that positively expansive epimorphisms are necessarily restricted to finite systems. Finally, we demonstrate a robust connection with topological dynamics via Pontryagin duality: algebraic expansivity on torsion abelian groups is shown to be exactly the dual property of topological expansivity on totally disconnected compact groups.

math.DS↗

arXiv2023

Abstract entropy and expansiveness

We present a general definition of entropy in the setting of pre-ordered semigroups, extending the notion of topological entropy. From our definition, we obtain the basic properties exhibited by various entropy-like theories encountered in the literature, many of them being particular cases of our scheme. We show how to derive from the properties of the abstract entropy corresponding properties in the concrete cases. We also introduce a notion of expansiveness in this general context, extending the concept of expansive dynamical system, which is related to the entropy in a similar fashion as in the case of topological dynamics. Finally, as an application, we suggest some definitions of new entropies and expansiveness in concrete cases.

math.DS↗

arXiv2020

Extensions of Expansive Dynamical Systems

We characterize and describe the extensions of expansive and Anosov homeomorphisms on compact spaces. As an application we obtain a stability result for extensions of Anosov systems, and show a construction that embeds any expansive system inside an expansive system having the shadowing property for the pseudo orbits of the original space.

math.DS↗

arXiv2019

New cw-expansive homeomorphisms of surfaces

In this article we characterize monotone extensions of cw-expansive homeomorphisms of compact metric spaces. We study the topology of its quotient space in the case of a compact surface. These results are applied to prove that there are cw-expansive homeomorphisms of compact surfaces with infinitely many fixed points and empty wandering set. The examples are quotients of topological perturbations of pseudo-Anosov diffeomorphisms. We also show that there is a cw-expansive homeomorphism with the shadowing property of the 2-sphere.

math.DS↗

arXiv2019

Expansive systems on lattices

We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations of Mañé's Theorem and Utz's Theorem about the finite dimensionality of the phase space of an expansive system, and the finiteness of a space supporting a positively expansive homeomorphism, respectively. We also discuss the notion of entropy in this setting calculating it for the non-Hausdorff shifts.

math.DS↗

arXiv2019

A note on Anosov homeomorphisms

For an $α$-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the $α$-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.

math.DS↗

arXiv2016

Observing expansive maps

We consider the problem of the observability of positively expansive maps by the time series associated to continuous real functions. For this purpose we prove a general result on the generic observability of a locally injective map of a compact metric space of finite topological dimension, extending earlier work by Gutman [6]. We apply this result to partially solve the problem of finding the minimal number of functions needed to observe a positively expansive map. We prove that two functions are necessary and sufficient for positively expansive maps on tori.

math.DS↗

arXiv2016

Green's theorem for crossed products by Hilbert $C^*$-bimodules

Green's theorem gives a Morita equivalence $C_0(G/H,A)\rtimes G\sim A\rtimes H$ for a closed subgroup $H$ of a locally compact group $G$ acting on a $C^*$-algebra $A$. We prove an analogue of Green's theorem in the case $G=\mathbb{Z}$, where the automorphism generating the action is replaced by a Hilbert $C^*$-bimodule.

math.OA↗

arXiv2015

Expansive Homeomorphisms on non-Hausdorff Spaces

We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological properties are studied in relation to separability axioms, metrizability and uniform expansivity.

math.GN↗

arXiv2014

The Stable Rank of C*-modules

We prove equality between the Topological Stable Rank and the Bass Stable Rank for finitely generated projective left modules over a unital C*-algebra. In order to do so, the concept of Stable Rank of a Hilbert module is introduced.

math.OA↗

arXiv2005

Cuntz-Pimsner C-algebras and crossed products by Hilbert C-bimodules

Given a correspondence X over a C*-algebra A, we construct a C*-algebra and a Hilbert C*-bimodule over it whose crossed product is isomorphic to the augmented Cuntz-Pimsner C*-algebra of X. This construction enables us to establish a condition for two augmented Cuntz-Pimsner C*-algebras to be Morita equivalent.

math.OA↗