Ordinary Semicascades and Their Ergodic Properties

A relationship is considered between ergodic properties of a discrete dynamical system on a compact metric space $\Omega$ and characteristics of companion algebro-topological objects, namely, the Ellis enveloping semigroup $E$, the Köhler enveloping operator semigroup $\Gamma$, and the semigroup $G$ being the closure of the convex hull of $\Gamma$ in the weak-star topology on the operator space $\operatorname{End}C^*(\Omega)$. The main results are formulated for ordinary (having metrizable semigroup $E$) semicascades and for tame dynamical systems determined by the condition $\operatorname{card}E\le\mathfrak c$. A classification of compact semicascades in terms of topological properties of the semigroups specified above is given.