Symbolic extensions in intermediate smoothness on surfaces (Q2903106)

Let \(T:M\rightarrow M\) be a \(C^r\) map, \(r>1\), on a compact manifold. We say that \(T\) admits symbolic extensions if the shift map \(\sigma :Y\rightarrow Y\), where \(Y\) is a closed subset of \(\{ 0,1,\dots,M-1\} ^{Z}\), \(M>1\), is an extension of \(T\), i.e., \(T\) is a factor of \(\sigma\). The symbolic extension is then the infimum of the topological entropy on all the possible symbolic extensions of \(T\). The author solves a conjecture by Downarowicz and Newhouse for compact surfaces and \(C^r\) maps, \(r>1\). The existence for this kind of maps is proved and the symbolic extension entropy is proved to be bounded by \(h(T)+4R(T)/(r-1)\), where \(h(T)\) is the topological entropy of \(T\) and \(R(T)=\lim _n \log \|DT^n\|/n\). The inequality can be sharpened by local surface diffeomorphisms by \(h(T)+R(T)/(r-1)\).