Markov partitions for Anosov diffeomorphisms. (Particiones de Markov para difeomorfismos de Anosov.) (Spanish)

TL;DR

This paper provides an accessible introduction to Markov partitions for Anosov diffeomorphisms, explaining Sinai's theorem, Bowen's constructive method, and their relation to symbolic dynamics for students in mathematics.

Contribution

It revisits and explains the classic theory of Markov partitions for Anosov diffeomorphisms, emphasizing the constructive proof and its connection to symbolic dynamics.

Findings

Existence of Markov partitions with arbitrarily small diameter for Anosov diffeomorphisms.

Restatement of Bowen's constructive method for proving Sinai's theorem.

Relation of invariant submanifolds and semiconjugation to Bernoulli shifts.

Abstract

This monographic short book is intended to a brief introduction in the classic topic of the theory of Hiperbolic Dynamical Systems, for Spanish speaking students of an undergraduate course in Mathematics. We revisit the classic definition of Markov Partitions for Anosov diffeomorphisms on compact manifolds. We focus on the theorem of Sinai (Func. Anal. and its Applic. 2, 1968) stsating the existence of such partitions with arbitrarily small diameter. To prove the theorem of Sinai we restate the constructive method in Bowen, Lecture Notes in Math. 470, 1975. Finally we also relate the results, the method of their proofs and the properties of the invariant submanifolds, with the semiconjugation of the Anosov diffeomorphism with a Bernoulli shift, via the symbolic dynamics.