Markov partitions for hyperbolic sets
Let \(M\) be a compact manifold without boundary and let \(\text{Diff}(M)\) denote the set of diffeomorphisms from \(M\) to itself. Let \(f \in \text{Diff}(M)\) be given and \(\Lambda \subset M\) be a hyperbolic set for \(f\). If \(V\) is a neighborhood of \(\Lambda\) then it was shown by the first author of this paper [Ergodic Theory Dyn. Syst. 26, No.~5, 1491--1509 (2006; Zbl 1122.37022)] that there exists a hyperbolic set \({\widetilde \Lambda}\) for \(f\) such that \(\Lambda \subset {\widetilde \Lambda}\subset V\) and \({\widetilde \Lambda}\) has a Markov partition. Then there is a canonically associated symbolic space called a subshift of finite type associated to the Markov partition. The dynamical systems of \(f\) on \({\widetilde \Lambda}\) can be studied by the method of symbolic dynamics. Recall that a dynamical system \((X, f)\) is called topologically mixing if for any open sets \(U\) and \(V\) there exists some positive integer \(N\) such that \(f^n(U)\cap V \neq \emptyset\) for all \(n\geq N\); the dynamical system is topologically transitive if there exists a point \(x\in X\) such that the forward orbit of \(x\) (with respect to \(f\)) is dense in \(X.\) The purpose of this paper under review is to strengthen the result on Markov partition in the paper mentioned above. If \(\Lambda\) is a topologically mixing hyperbolic set for \(f\in \text{Diff}(M)\) and \(V\) is a neighborhood of of \(\Lambda\), then the authors prove that there exists a hyperbolic set \({\widetilde \Lambda}\) for \(f\) containing \(\Lambda\) and contained in \(V\) such that \(({\widetilde \Lambda}, f)\) has a Markov partition coming from an associated mixing subshift of finite type. Furthermore, if \((\Lambda, f)\) is transitive, then \(({\widetilde \Lambda}, f)\) has a Markov partition coming from an associated transitive subshift of finite type. As a corollary of their main result they show that with the same datum \((\Lambda, f)\) and \(V\) as in the main theorem, there exists a unique measure of maximal entropy (on \({\widetilde \Lambda}\)) with respect to \(f\).
scientific article; zbMATH DE number 5706079