On certain hyperbolic sets
A hyperbolic invariant set \(F\) is called ``locally maximal'' under the action of a diffeomorphism \(S\) if there exists a neighborhood \(U\) of \(F\) such that the maximal \(S\)-invariant subset in \(U\) is \(F\). An invariant, not locally maximal set \(F\) is called ``locally premaximal'' if any neighborhood of \(F\) contains a locally maximal set which contains \(F\). There are two examples of hyperbolic sets which are not locally premaximal, due to \textit{S. Crovisier} [C. R., Math., Acad. Sci. Paris 334, No. 5, 401--404 (2002; Zbl 1042.37022)] and due to \textit{T. Fisher} [Ergodic Theory Dyn. Syst. 26, No. 5, 1491--1509 (2006; Zbl 1122.37022)]. The author shows that, for some hyperbolic set \(F'\) to be not locally premaximal, it is sufficient to be equihomeomorphic to the hyperbolic sets either from Crovisier's and from Fisher's example. This is an interesting assertion, since local premaximality (or its absence) is a property referring to the behavior of orbits off, but near \(F'\), yet equihomeomorphisms relate only the intrinsic dynamics.
scientific article; zbMATH DE number 6209065