Shadowing and internal chain transitivity

Abstract

The main result of this paper is that a map f : X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with shadowing have the property that every internally chain transitive set is an ω-limit set of a point, and we also show that topologically hyperbolic maps and certain quadratic Julia sets have a closed space of ω-limit sets 2000 Mathematics Subject Classification. 37B50, 37B10, 37B20, 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]).…”

“…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].…”

Shadowing, asymptotic shadowing and s-limit shadowing

“…The answer to [BDG12, Conjecture 1.2] came from Meddaugh and Raines in [MR13]. They showed that the set ICT (f ) must be closed in 2 X , the hyperspace of closed non-empty subsets of X furnished with the Hausdorff distance.…”

Orbital shadowing, internal chain transitivity and -limit sets

“…However, in [15], the authors prove that, under the assumption that f has the shadowing property, ω f is closed precisely when it is equal to the set of internally chain transitive sets of f . The paper demonstrates this by effectively proving the following two results (the second of which is only implicitly proven in the original paper.)…”

“…More recently, it has been demonstrated that for systems with the shadowing property, ω-limit sets are completely characterized by internal chain transitivity if and only if ω f is closed with respect to the Hausdorff topology [15]. A map f : X → X has the shadowing property provided that for all > 0 there exists a δ > 0 such that for any δ-pseudo-orbit x i (i.e.…”