This dissertation addresses the following three topics relating to the structure of hyperbolic sets: (1) hyperbolic sets that are not contained in locally maximal hyperbolic sets; (2) the existence of a Markov partition for a hyperbolic set; (3) and hyperbolic sets which contain nonempty interior.
In Chapter 3 we construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The examples are robust under perturbations and can be built on any compact manifold of dimension greater than one.
In Chapter 4 we show that every hyperbolic set is included in a hyperbolic set with a Markov partition. Also, we describe a condition that ensures a hyperbolic set is included in a locally maximal hyperbolic set.
In Chapter 5 we construct two further examples of hyperbolic sets that are not contained in any locally maximal hyperbolic set. The first example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The second example is symplectic.
In Chapter 6 we study hyperbolic sets with nonempty interior. We prove the folklore theorem that every transitive hyperbolic set with interior is Anosov. We also show that on a compact surface every locally maximal set with nonempty interior is Anosov. Finally, we give examples of hyperbolic sets with nonempty interior for a non-Anosov diffeomorphism.