Symbolic extensions in intermediate smoothness on surfaces

Abstract

We prove that $C^r$ 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.