Symbolic extensions in intermediate smoothness on surfaces

TL;DR

This paper proves that smooth maps of class greater than one on compact surfaces admit symbolic extensions, providing a precise bound on the symbolic extension entropy, thus confirming a conjecture and advancing understanding in dynamical systems.

Contribution

It establishes the existence of symbolic extensions for $ ext{C}^r$ maps on surfaces with a sharp entropy bound, confirming a conjecture and improving prior results.

Findings

Existence of symbolic extensions for $ ext{C}^r$ maps with $r>1$ on surfaces.

A sharp upper bound on symbolic extension entropy is provided.

The conjecture of S. Newhouse and T. Downarowicz is confirmed in dimension two.

Abstract

We prove that $\mathcal{C}^r$ maps with $r>1$ on a compact surface have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S.Newhouse and T.Downarowicz in dimension two and improves a previous result of the author \cite{burinv}.