  We establish two results under which the topology of a hyperbolic set
constrains ambient dynamics. First if a set is a compact, transitive, expanding
hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a
union of transitive, expanding attractors (or contracting repellers) of
codimension 1 for any diffeomorphism such that it is hyperbolic. Secondly, if a
set is a nonwandering, locally maximal, compact hyperbolic set for a surface
diffeomorphism, then it is locally maximal for any diffeomorphism for which it
is hyperbolic.
