On Certain Hyperbolic Sets

Two invariant sets $F$ of certain diffeomorphisms $S$ that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of hyperbolic sets with the property (unexpected at that time) that, in some neighborhood of such an $F$, there is no locally maximal set containing $F$ are considered. It is proved that this property, although referring to the behavior of the orbits of $S$ near $F$, is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of $S$ on $F$ with the hyperbolicity of $F$. (This means that if a hyperbolic set $F'$ for a diffeomorphism $S'$ is equivariantly homeomorphic to a Fathi–Crovisier or a Fisher set, then $F'$ has a neighborhood in which $S'$ has no locally maximal set containing $F'$.)