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Regular and Chaotic Dynamics, 2024, Volume 29, Issue 1, paper published in the English version journal
DOI: https://doi.org/10.1134/S1560354724010118
(Mi rcd1252)
 

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
(1)
References:
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
Funding agency Grant number
Russian Science Foundation 22-21-00304
Ministry of Science and Higher Education of the Russian Federation 075- 15-2022-1101
This work was supported by the Russian Science Foundation (project No 22-21-00304), except Section 4, whose results were supported by the Laboratory of Dynamical Systems and Applications NRU HSE, by the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2022-1101).
Received: 21.07.2023
Accepted: 22.12.2023
Document Type: Article
MSC: 54H200, 37B05
Language: English
Citation: Nina I. Zhukova
Citation in format AMSBIB
\Bibitem{Zhu24}
\by Nina I. Zhukova
\mathnet{http://mi.mathnet.ru/rcd1252}
\crossref{https://doi.org/10.1134/S1560354724010118}
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