A characterization ofω-limit sets in shift spaces

Abstract

A set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z∈X with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

On the ω-limit sets of tent maps

“…For an interval map f : I → I, a subinterval J ⊂ I is called a homterval if c / ∈ f n (J) for every n ≥ 0 and any local extremum c. Notice that tent maps T with slope λ ∈ (1, 2] have no homterval, since every subinterval expands under T until eventually it contains the critical point c. Lemmas 2.4, 2.5 and 2.6 are well-known [5,6,16,18]. We state them here as they will be of use in what follows.…”

“…Furthermore, they have been the subject of many research articles [12,17,28,30], often in relation to their ω-limit sets [6,21,22,23,24]. In this paper we make several important observations about the behaviour of tent maps, allowing us to prove new results about the nature of their limit sets in relation to certain well-known dynamical properties.…”

“…One of the first papers to link ω-limit sets to the property of internal chain transitivity was [26], in which Hirsch et al show that every ω-limit set is internally chain transitive, and furthermore every compact, internally chain transitive set is the ω-limit set of some asymptotic pseudo-orbit. We have followed up this work in several articles: in [6] we show that for shifts of finite type, all closed, internally chain transitive sets are ω-limit sets; in [5] we show that for interval maps containing no homtervals (i.e. maps whose pre-critical points are dense in the interval, such as tent maps), all closed, internally chain transitive sets that do not contain the image of a critical point are ω-limit sets; in [7] we show that for the full tent map (that with slope equal to 2), all internally chain transitive sets are ω-limit sets.…”

Shadowing and internal chain transitivity

“…It is known that there are interval maps both with and without shadowing, and partial classifications exist in this context [11]. It has also been demonstrated that shifts of finite type [5] and Julia sets for certain quadratic maps [7,8] all exhibit shadowing .…”

“…Specifically, for several types of dynamical systems it is known that ( †) A closed set A is internally chain transitive if, and only if, there is some x with ω(x) = A. 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.…”