  For a continuous semicascade on a metrizable compact set $\Omega $, we
consider the weak$^{*}$ convergence of generalized operator ergodic means in
${\rm End}\, \, C^{*} (\Omega)$. We discuss conditions on the dynamical system
under which (a) every ergodic net contains a convergent subsequence; (b) all
ergodic nets converge; (c) all ergodic sequences converge. We study the
relationships between the convergence of ergodic means and the properties of
transitivity of the proximality relation on $\Omega$, minimality of supports of
ergodic measures, and uniqueness of minimal sets in the closure of trajectories
of a semicascade.
  These problems are solved in terms of three algebraic-topological objects
associated with the dynamical system: the Ellis enveloping semigroup, the
K\"{o}hler operator semigroup $\Gamma $, and the semigroup $G$ that is the
weak$^{*} $ closure of the convex hull of $\Gamma $ in ${\rm End}\, C^{*}
(\Omega)$. The main results are stated for ordinary semicascades (whose Ellis
semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary
is equivalent to being "nonchaotic" in an appropriate sense. We present a
classification of compact dynamical systems in terms of topological properties
of the above-mentioned semigroups.

Dynamical Systems