Nina Zhukova

Continuous actions of topological semigroups on products $X$ of an arbitrary family of topological spaces $X_i$, $i\in J,$ are studied. The relationship between the dynamical properties of semigroups acting on the factors $X_i$ and the same properties of the product of semigroups on the product $X$ of these spaces is investigated. We consider the following dynamical properties: topological transitivity, existence of a dense orbit, density of a union of minimal sets, and density of the set of points with closed orbits. The sensitive dependence on initial conditions is investigated for countable products of metric spaces. Various examples are constructed. In particular, on an infinite-dimen\-sio\-nal torus we have constructed a continual family of chaotic semi\-group dynamical systems that are pairwise topologi\-cal\-ly not conjugate by homeomorphisms preserving the structure of the product of this torus.

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.