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′.)