Teor\'ia Erg\'odica de los Atractores Topol\'ogicos y Estad\'isticos (Ergodic Theory of Topological and Statistical Attractors) (Spanish)

TL;DR

This textbook introduces graduate students to ergodic properties of topological and statistical attractors, covering foundational concepts and advanced theories including SRB measures, Gibbs measures, and statistical attractors in dynamical systems.

Contribution

It provides a comprehensive, accessible overview of ergodic theory and attractors, integrating classical and recent developments for systems with varying differentiability.

Findings

Review of basic ergodic concepts and properties

Development of theory for topological and statistical attractors

Discussion of existence and relations of SRB and physical measures

Abstract

This is a textbook, in Spanish, for an introductory short course that focusses on the ergodic properties of topological and statistical attractors. It is directed to graduate students in Mathematics and related Science, in particular to the students-participants of the XXVI Escuela Venezolana de Matem\'atica. The first three chapters include a review of the basic concepts and properties of the ergodic theory and of the topological and differentiable dynamics of systems that evolve on discrete time. They do not require from the reader previous knowledge on Dynamical Systems. The latter three chapters develop the theory of topological attractors, SRB measures, and the existence of Gibbs and physical measures for hyperbolic and sufficiently differentiable systems. Finally, for continuous and non necessarily differentiable systems, we include a review of Milnor's and Ilyashenko's or statistical attractors, the proof of their existence and the relation between statistical attractors and the "SRB-like" or pseudo-physical measures.