Nina Zhukova
B. Pecherskaja, 25/12, Nizhny Novgorod, 603155, Ru
National Research University Higher School of Econ
Publications:
Zhukova N. I.
Sensitivity and Chaoticity of Some Classes of Semigroup Actions
2024, vol. 29, no. 1, pp. 174189
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 abovementioned 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 wellknown 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.
