Symbolic dynamics for certain non-invertible $C^{1+\beta}$ maps

TL;DR

This paper develops a symbolic dynamics framework for certain non-invertible $C^{1+eta}$ maps with zero Lyapunov exponents and singularities, using recent techniques to construct Markov partitions and symbolic extensions.

Contribution

It introduces a method to construct countable Markov partitions and finite-to-one symbolic extensions for non-invertible maps with singularities, advancing the understanding of their dynamical structure.

Findings

Constructed a countable Markov partition for the invariant set.

Established the existence of a finite-to-one symbolic extension.

Extended symbolic dynamics techniques to maps with zero Lyapunov exponents and singularities.

Abstract

Let $f$ be a non-invertible $C^{1+\beta}(\beta>0)$ map with zero Lyapunov exponents and singularities on a closed Riemannian manifold $M$. We consider the symbolic dynamics of $f$. Combining the techniques in recent works of Sarig, Ovadia and Araujo-Lima-Poletti, we construct a countable Markov partition for the invariant set consisting of summable points of the inverse limit space of $(M, f)$ and show that there exists a finite-to-one symbolic extension for $f$ on the corresponding subset of $M$.