Sensitivity and Chaoticity of Some Classes of Semigroup Actions

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.