Characterizations of $\omega$-limit sets in topologically hyperbolic systems

Abstract

It is well known that ω-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 ω-limit set, and separately that in shifts of finite type, a set is internally chain transitive if and only if it is a (regular) ω-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.2000 Mathematics Subject Classification. 37B25, 37B45, 37E05, 54F15, 54H20.

Preservation of shadowing in discrete dynamical systems

“…As mentioned in the introduction, Bowen [6] was one of the first to us the property of shadowing in his study of Axiom A diffeomorphisms and since then it has been both used as a tool and studied extensively in a property in its own right (see, for examples, [2,7,11,12,13,20,27,29,31,34,35,37,39,40,41]).…”

“…Condition (1) simply says that part of what it means for a system to have s-limit shadowing is that it has shadowing. Suppose that (X, f ) satisfies condition (2). Let E ∈ U be given and take a corresponding D ∈ U .…”

“…Since our space is compact Hausdorff throughout this paper, it follows from Theorem 11.0.1 that when checking for s-limit shadowing it suffices to verify whether or not condition (2) in the definition holds. 11.1.…”

“…Whilst shadowing is clearly important when modelling a system numerically (for example [11,34]), it is also been found to have theoretical importance; for example, Bowen [6] used shadowing implicitly as a key step in his proof that the nonwandering set of an Axiom A diffeomorphism is a factor of a shift of finite type. Since then it has been studied extensively, in the setting of numerical analysis [11,12,34], as an important factor in stability theory [37,39,41], in understanding the structure of ω-limit sets and Julia sets [2,3,4,7,29], and as a property in and of itself [13,20,27,31,35,37,40].…”

On the shadowing and limit shadowing properties

“…Also in [25], it is proved that for circle homeomorphisms, the shadowing property always implies the limit shadowing property, and the same implication holds true for c-expansive maps including expansive homeomorphisms (see [4,5,17]). It is rather difficult to construct a continuous map satisfying the shadowing property but not the limit shadowing property, but in [12], such an example is given, while the equivalence of the two shadowing properties is proved for a certain class of interval maps.…”