Shadowing and -limit sets of circular Julia sets

Abstract

Abstract. We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

Preservation of shadowing in discrete dynamical systems

“…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].…”

Distributional chaos in dendritic and circular Julia sets

“…By Lemma 2.12 in [4], for any N ∈ N, there exists a δ N such that d(x, y) < δ N implies x N y N . Take N = 1 and recall that x 1 has length 2.…”

“…By Lemma 2.12 in [4], there exists N ∈ N such that if x N y N , then d(x, y) < . Certainly, then, x N = y N implies d(x, y) < , so…”

“…, meaning there exists a subsequence μ j such that α μ j = β μ j , so that the words u(α μ j , μ j ) and u(β μ j , μ j ) disagree in every place. By Theorem 22 in [3], for any N ∈ N, there exists a δ N such that d(x, y) < δ N implies x N ∼ y N . Take N = 1.…”

Shadowing and internal chain transitivity

“…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.…”