Dynamical Properties of Continuous Semigroup Actions and Their Products

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-dimensional torus we have constructed a continual family of chaotic semigroup dynamical systems that are pairwise topologically not conjugate by homeomorphisms preserving the structure of the product of this torus.