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Special Issue: In Honor of Vladimir Belykh and Sergey Gonchenko Guest Editors: Alexey Kazakov, Vladimir Nekorkin, and Dmitry Turaev
Sensitivity and Chaoticity of Some Classes of Semigroup Actions
Nina I. Zhukova HSE University,
ul. Bolshaja Pecherskaja 25/12, 603155 Nizhny Novgorod, Russia
Abstract:
The focus of the work is the investigation of chaos and closely related dynamic properties of continuous actions of almost open
semigroups and $C$-semigroups. The class of dynamical systems $(S, X)$ defined by such semigroups $S$ is denoted by $\mathfrak A$.
These semigroups contain, in particular, cascades, semiflows and groups of homeomorphisms. We extend the Devaney definition of chaos to general dynamical systems. For $(S, X)\in\mathfrak A$ on locally compact metric spaces $X$ with a countable base we
prove that topological transitivity and density of the set formed by points having closed orbits imply the sensitivity to initial conditions. We assume neither the compactness of metric space nor the compactness of the above-mentioned closed orbits.
In the case when the set of points having compact orbits is dense, our proof proceeds without the assumption of local compactness of the phase space $X$. This statement generalizes the well-known result of J. Banks et al. on Devaney's definition
of chaos for cascades.The interrelation of sensitivity, transitivity and the property of minimal sets of semigroups is investigated. Various examples are given.
Keywords:
semigroup, topological transitivity, chaotic semigroup, minimal set, sensitivity
Received: 21.07.2023 Accepted: 22.12.2023
Citation:
Nina I. Zhukova
Linking options:
https://www.mathnet.ru/eng/rcd1252
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Abstract page: | 34 | References: | 20 |
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