$\mathcal{C}^2$ surface diffeomorphisms have symbolic extensions

TL;DR

This paper proves that all $ ext{C}^2$ surface diffeomorphisms can be represented by symbolic systems, using a novel approach that combines hyperbolic theory and Yomdin's theory to bound local entropy via Lyapunov exponents.

Contribution

It establishes the existence of symbolic extensions for $ ext{C}^2$ surface diffeomorphisms by bounding local entropy through a new reparametrization method.

Findings

Bounded local entropy in terms of Lyapunov exponents.

Established symbolic extensions for all $ ext{C}^2$ surface diffeomorphisms.

Combined hyperbolic and Yomdin's theories for entropy analysis.

Abstract

We prove that $\mathcal{C}^2$ surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. Following the strategy of T.Downarowicz and A.Maass \cite{Dow} we bound the local entropy of ergodic measures in terms of Lyapunov exponents. This is done by reparametrizing Bowen balls by contracting maps in a approach combining hyperbolic theory and Yomdin's theory.