Topological dynamics on hyperspaces

Sharma, Puneet; Nagar, Anima

doi:10.4995/agt.2010.1724

Cited by 19 publications

(19 citation statements)

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“…A special case when the family F is a singleton has been investigated by several authors and a lot of work in this direction has already been done [1,8,11,13,14,16]. It was proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace [1,16].…”

mentioning

confidence: 99%

“…Thus, it is important to study the set-valued dynamics induced by a continuous self map which inturn can help characterizing the dynamics of a general dynamical system. Many researchers have addressed the problem and many of the questions in this direction have been answered [1,8,11,13,14,16]. In the process, the dynamical behavior of a system and its corresponding set-valued counterpart has been investigated and several interesting results have been obtained.…”

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confidence: 99%

“…Interesting results relating the topological entropy of the two spaces have been obtained [8]. In [13], Sharma and Nagar investigated some of the natural questions arising from this setting. They investigated the influence of each of the individual units and the role of the underlying hyperspace topology in determining the dynamics induced by a continuous self map on a general topological space.…”

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confidence: 99%

“…It was proved that while weakly mixing and topological mixing on the two spaces are equivalent, transitivity on the base space need not imply transitivity on the hyperspace [1,16]. While [8] investigated the topological entropy of induced system of all nonempty compact subsets of X, [13] investigated the problem for a general hyperspace endowed with a general hyperspace topology. They discussed various dynamical notions like dense periodicity, transitivity, weakly mixing, topological mixing and topological entropy.…”

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confidence: 99%

“…They also investigated notions like sensitive dependence on initial conditions, topological entropy, Li-Yorke chaos, existence of Li-Yorke pairs, existence of horseshoe and the corresponding results were established. Investigating the inverse of the problem stated, they also investigated the behavior of an individual component of the system, given the dynamical behavior of the induced system on the hyperspace [13,14].…”

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confidence: 99%

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Induced dynamics on the hyperspaces

Sharma

2016

Appl. Gen. Topol.

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In this paper, we study the dynamics induced by finite commutative relation. We prove that the dynamics generated by such a non-trivial collection cannot be transitive/super-transitive and hence cannot exhibit higher degrees of mixing. As a consequence we establish that the dynamics induced by such a collection on the hyperspace endowed with any admissible hit and miss topology cannot be transitive and hence cannot exhibit any form of mixing. We also prove that if the system is generated by such a commutative collection, under suitable conditions the induced system cannot have dense set of periodic points. In the end we give example to show that the induced dynamics in this case may or may not be sensitive.

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