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Sensitivity and Chaoticity of Some Classes of Semigroup Actions
Regular and Chaotic Dynamics ( IF 1.4 ) Pub Date : 2024-03-11 , DOI: 10.1134/s1560354724010118
Nina I. Zhukova

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.



中文翻译:

某些类半群行为的敏感性和混沌性

该工作的重点是研究几乎开半群和\(C\) -半群的连续作用的混沌和密切相关的动态性质。由此类半群\ (S\ ) 定义的动力系统类\ ((S,X)\)表示为\(\mathfrak{A}\)。这些半群特别包含级联、半流和同态群。我们将德瓦尼对混沌的定义扩展到一般动力系统。对于具有可数基的局部紧度量空间\(X\)上的\((S,X)\in\mathfrak{A}\),我们证明由具有闭轨道的点形成的集合的拓扑传递性和密度意味着敏感性到初始条件。我们既不假设度量空间的紧致性,也不假设上述闭轨道的紧致性。在具有紧致轨道的点集密集的情况下,我们的证明在不假设相空间\(X\)局部紧致的情况下进行。该声明概括了 J. Banks 等人众所周知的结果。关于级联混沌的德瓦尼定义。研究了敏感性、传递性和半群最小集性质的相互关系。给出了各种例子。

更新日期:2024-03-11
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