Stable sets and \(\epsilon\)-stable sets in positive-entropy systems (Q2517921)

Let \(X\) be a compact metric space and let \(T:X\rightarrow X\) be a homeomorphism. For \(x\in X\), denote by \(W^s(x,T)\) its stable set and if \(\epsilon >0\), denote its \(\epsilon\)-stable set \(W^s_{\epsilon}(x,T)\) as the set of points whose forward orbit \(\epsilon\)-shadows the orbit of \(x\). The author proves that if the topological entropy is positive, it can be computed in terms of \((n,\delta)\)-separated subsets contained in \(T^{-n}W^s_{\epsilon}(x,T)\), being \(\delta >0\) and \(n\) a positive integer. With the additional assumption that \(X\) has finite covering dimension, a similar result was previously proved in [\textit{D. Fiebig, U.-R. Fiebig} and \textit{Z. H. Nitecki}, Ergodic Theory Dyn. Syst. 23, No. 6, 1785--1806 (2003; Zbl 1063.37017)].