A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology

Wang, Yangeng; Wei, Guo; Campbell, William H.; Bourquin, Steven

doi:10.1016/j.chaos.2008.07.014

Cited by 12 publications

(7 citation statements)

References 28 publications

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“…Blokh (see [35]) and Vellekoop and Berglund [37] show that, on the interval, transitivity implies Devaney chaos. Wang et al [39] state that non-metric spaces do not admit a notion of sensitivity. In [38], Wang et al consider various versions of Devaney chaos in the light of various notions related to transitivity.…”

Section: Chris Good and Sergio Macíasmentioning

confidence: 99%

What is topological about topological dynamics?

Good¹,

Macı́as²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric in compact metric spaces. We show for example that in a Tychonoff space, transitivity and dense periodic points imply (uniform) sensitivity to initial conditions. We generalise Bryant's result that a compact Hausdorff space admitting a c-expansive homeomorphism in the obvious uniform sense is metrizable. We study versions of shadowing, generalising a number of well-known results to the topological setting, and internal chain transitivity, showing for example that ω-limit sets are (uniform) internally chain transitive and weak incompressibility is equivalent to (uniform) internal chain transitivity in compact spaces.

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Section: Chris Good and Sergio Macíasmentioning

confidence: 99%

What is topological about topological dynamics?

Good¹,

Macı́as²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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“…Notice that in order to explore whether a group G induces a continuous action on the hyperspace CL(X) of a G-space X, we first need to solve a simpler question: whether a continuous map f : X → Y induces a continuous function f : CL(X) → CL(Y ) (see equation (3.1) for the precise definition of f ). Let us observe that in some contexts (such as dynamical systems [7,27]) the induced function plays an important role, and therefore this problem has been long studied (see e.g. [15,20,27]).…”

Section: Introductionmentioning

confidence: 99%

“…Let us observe that in some contexts (such as dynamical systems [7,27]) the induced function plays an important role, and therefore this problem has been long studied (see e.g. [15,20,27]). Until now, the continuity of f is clear if we endow the hyperspace CL(X) with the Vietoris or the Hausdorff metric topology.…”

Section: Introductionmentioning

confidence: 99%

Some notes on induced functions and group actions on hyperspaces

Donjuán¹,

Jonard-Pérez²,

López-Poo³

2021

Preprint

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Let X be a topological space and CL(X) be the family of all nonempty closed subsets of X.In this paper we discuss the problem of when a continuous map between topological spaces induces a continuous function between their respective hyperspaces. As a main result we characterize the continuity of the induced function in the case of the Fell and Attouch-Wets hyperspaces. Additionally we explore the problem of whether a continuous action of a topological group G on a topological space X induces a continuous action on CL(X). In particular we give sufficient conditions on the topology of G to guarantee that the induced action on CL(X) is continuous, provided that CL(X) is equipped with the Hausdorff or the Attouch-Wets metric topology.

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“…It is well-known that (E, f) is topologically conjugated to the subsystem of (2 E , 2 f ) that consists of the singleton sets of E when E satisfies certain conditions and an appropriate hyperspace topology is selected, e.g., hit-or-miss topology (see [28][29][30]) or Vietoris topology (see [3,16,36]). Clearly, an invariant subset of the original system (E, f) becomes a fixed point of the hyperspace system (2 E , 2 f ).…”

Section: Introductionmentioning

confidence: 99%