Uncountable ω-limit sets with isolated points

Good, Chris; Raines, Brian E.; Suabedissen, Rolf

doi:10.4064/fm205-2-6

Cited by 3 publications

(7 citation statements)

References 9 publications

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“…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.…”

Section: Introductionmentioning

confidence: 84%

“…Many recent results on ω-limit sets of interval maps have used symbolic dynamics and kneading theory (see [5,6,23,24] for examples). In this paper we use results and techniques from symbolic dynamics and kneading theory, together with conventional analysis of interval maps, extending the theory in both areas where necessary.…”

Section: Introductionmentioning

confidence: 99%

“…It is known that ω-limit sets are non-empty, closed and invariant (by which we mean that f (ω(x)) = ω(x)). They have been studied extensively, with particular focus on the ω-limit sets of interval maps [1,4,5,6,7,9,10,11,14,15,21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

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On the ω-limit sets of tent maps

2012

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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.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

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On the ω-limit sets of tent maps

2012

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“…Indeed, all the points of S are isolated (non-periodic) recurrent points, whereas for continuous maps such points are necessarily periodic. On the other hand, for a continuous map defined on a compact set X, if a point x ∈ X satisfies ω(x) = S ∪ K, where K is a minimal Cantor set and S is a scattered subset of ω(x), then S is dense in ω(x) [12]. Example 4.3 shows also that this result does not hold for piecewise contracting maps.…”

Section: Dynamics On the Attractormentioning

confidence: 99%

“…Let (x, y, z) and (x , y , z ) be two different points of S n, 0 , and w, w be their respective corresponding word in {1, 2} n . Then, using (12) and (13) and applying the triangular inequality, we obtain that d(f p0k0 (x, y, z), f p0k0 (x , y , z )) > 0 , for a p 0 ∈ {1, . .…”

Section: Disconnectedness and Complexitymentioning

confidence: 99%

On the asymptotic properties of piecewise contracting maps

et al. 2015

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We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint open pieces. We focus on the topological and the dynamical properties of the (global) attractor of the orbits that remain in this union. As a starting point, we show that the attractor consists of a finite set of periodic points when it does not intersect the boundary of a contraction piece, which complements similar results proved for more specific classes of piecewise contracting maps. Then, we explore the case where the attractor intersects these boundaries by providing examples that show the rich phenomenology of these systems. Due to the discontinuities, the asymptotic behaviour is not always properly represented by the dynamics in the attractor. Hence, we introduce generalized orbits to describe the asymptotic dynamics and its recurrence and transitivity properties. Our examples include transitive and recurrent attractors, that are either finite, countable, or a disjoint union of a Cantor set and a countable set. We also show that the attractor of a piecewise contracting map is usually a Lebesgue measure-zero set, and we give conditions ensuring that it is totally disconnected. Finally, we provide an example of piecewise contracting map with positive topological entropy and whose attractor is an interval.

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

Barwell¹,

Good²,

Oprocha

et al. 2013

Discrete &Amp; Continuous Dynamical Systems - A

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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.

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Uncountable ω-limit sets with isolated points

Cited by 3 publications

References 9 publications

On the ω-limit sets of tent maps

On the ω-limit sets of tent maps

On the asymptotic properties of piecewise contracting maps

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

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