arXiv subjects

Túlio Vales

Publications and source records attributed to Túlio Vales.

4 recordsLinked to original sources

arXiv2025

Spectral decomposition and entropy for open set-valued maps

In this paper, we study the dynamics of set-valued maps whose graphs are open and such that the image of each point is an open and connected set. Building upon the work of P. Duarte and M. Torres, who introduced and analyzed the combinatorial structure of the final recurrent set, we incorporate the notions of transitivity and mixing, thereby bringing their framework into the spirit of Smale's spectral decomposition theorem. We also demonstrate that such maps exhibit infinite topological entropy, and we establish a connection between transitive open set-valued maps and transitive Anosov diffeomorphisms.

math.DS↗

arXiv2024

Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

Given a random map (T_1, T_2, T_3, T_4, p_1, p_2, p_3, p_4), we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.

math.DS↗

arXiv2022

A Random Billiard Map in the Circle

We consider a random billiard map, the one in which the standard specular reflection rule is replaced by a random reflection given by a Markov operator. We exhibit an invariant measure for random billiards on general tables. In the special case of a circular table we show that almost every (random) orbit is dense in the boundary as well as in the circular ring formed between the circle boundary and the random caustic. We additionally prove Strong Knudsen's Law for a particular case of families of absolutely continuous measures with respect to Liouville.

math.DS↗

arXiv2020

A Symmetric Random Walk defined by the Time-One Map of a Geodesic Flow

In this note we consider a symmetric random walk defined by a $(f,f^{-1})$ Kalikow type system, where $f$ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the random walk are deduced in these cases.

math.DS↗