  It is well known that \omega-limit sets are internally chain transitive and
have weak incompressibility; the converse is not generally true, in either
case. However, it has been shown that a set is weakly incompressible if and
only if it is an abstract \omega-limit set, and separately that in shifts of
finite type, a set is internally chain transitive if and only if it is a
(regular) \omega-limit set. In this paper we generalise these and other
results, proving that the characterization for shifts of finite type holds in a
variety of topologically hyperbolic systems (defined in terms of expansive and
shadowing properties), and also show that the notions of internal chain
transitivity and weak incompressibility coincide in compact metric spaces.
