Hyperbolic sets that are not locally maximal

Abstract

This paper addresses the following topics relating to the structure of hyperbolic sets: first, hyperbolic sets that are not contained in locally maximal hyperbolic sets; second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a four-dimensional manifold. The third example is volume-preserving and constructed on $\mathbb{R}^4$. We show that every hyperbolic set is included in a hyperbolic set with a Markov partition. In addition, we describe a condition that ensures a hyperbolic set is included in a locally maximal hyperbolic set.