Hyperbolic sets that are not contained in a locally maximal one
Résumé
In this paper we study two properties related to the structure of hyperbolic sets. First we construct new examples answering in the negative the following question posed by Katok and Hasselblatt in [B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems Cambridge University Press, 1995, p. 272].
Question. Let $\Lambda$ be a hyperbolic set, and let $V$ be an open neighborhood of $\Lambda$. Does there exist a locally maximal hyperbolic set $\widetilde{\Lambda}$ such that $\Lambda \subset \widetilde{\Lambda} \subset V$ ?
We show that such examples are present in linear Anosov diffeomorophisms of $\mathbb{T}^3$, and are therefore robust.
Also we construct new examples of sets that are not contained in any locally maximal hyperbolic set. The examples known until now were constructed by Crovisier in [S. Crovisier , Une remarque sur les ensembles hyperboliques localement maximaux, C. R. Math. Acad. Sci. Paris, 334 (2002) , 401-404] and by Fisher in [T. Fisher , Hyperbolic sets that are not locally maximal, Ergodic Theory and Dynamical Systems, 26 (2006) , 1491-1509], and these were either in dimension equal or bigger than 4 or they were not transitive. We give a transitive and robust example in $\mathbb{T}^3$. And show that such examples cannot be build in dimension 2.