Local maximality of hyperbolic sets

Two properties of a hyperbolic set $F$ are discussed: its local maximality and the property that, in any neighborhood $U\supset F$, there exists a locally maximal set $F'$ that contains $F$ (we suggest calling the latter property local premaximality). Although both these properties of the set $F$ are related to the behavior of trajectories outside $F$, it turns out that, in the class of hyperbolic sets, the presence or absence of these properties is determined by the interior dynamics on $F$.