On the ω-limit sets of tent maps

Abstract

Abstract. For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every δ > 0 there is a sequence of points x = x 0 , x 1 , . . . , x n = y such that d(f (x i ), x i+1 ) < δ for 0 ≤ i < n. It is known that every ω-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an ω-limit set for some point in X, and that the same is also true for the full tent map T 2 : [0, 1] → [0, 1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an ω-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an ω-limit set. Together, these results lead us to conjecture that for maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

Shadowing and expansivity in subspaces

“…By Remark 5.1, f is open on U and f is ball expanding on U , thus by Theorem 4.3, f has h-shadowing on U . Shadowing follows directly from h-shadowing, and Theorem 3.7 (2) gives us that f has s-limit shadowing on Λ, since Λ ⊆ U .…”

“…Bowen was one of the first to consider this property in [6], where he used it in the study of ω-limit sets of Axiom A diffeomorphisms. In [2], we use shadowing to characterize ω-limit sets of tent maps, and, following on from [3], in [4] we use various forms of shadowing to characterize ω-limit sets of topologically hyperbolic systems. Of particular interest is a property called h-shadowing, which we prove is equivalent to shadowing in certain expansive systems (such as shifts of finite type), but is in general a stronger property, and one which allows us to prove when internally chain transitive sets are necessarily ω-limit sets [4].…”

Shadowing and internal chain transitivity

“…In particular, this applies to interval maps, so that Conjecture 1.2 of [4] can be answered in the affirmative. In that same paper, the authors conjecture that if f : X → X is a dynamical system on a compact metric space with shadowing, then ω(f ) = ICT (f ).…”

“…While it is known that there are systems which do not exhibit this ω(f )-ICT (f ) equality [5], in all of these systems, shadowing is either absent or unknown. This has lead to the conjecture that in systems which exhibit shadowing, ICT (f ) = ω(f ) [4].…”

“…It is known that ( †) holds for shifts of finite type, topologically hyperbolic maps, a family of quadratic Julia sets, and certain interval maps, [5], [8], [7], and [3]. In fact, in [4], Barwell et al show that there are certain unimodal maps of the unit interval for which ( †) holds and certain other ones for which it fails. They conjecture that the key to ( †) on the interval is the property of shadowing (defined in the next section).…”