  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. 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 and by Fisher, and these were either in dimension 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.
