  We study the problem on the weak-star decomposability of a topological
$\mathbb{N}_{0}$-dynamical system $(\Omega,\varphi)$, where $\varphi$ is an
endomorphism of a metric compact set $\Omega$, into ergodic components in terms
of the associated enveloping semigroups. In the tame case (where the Ellis
semigroup $E(\Omega,\varphi)$ consists of $B_{1}$-transformations
$\Omega\rightarrow \Omega$), we show that (i) the desired decomposition exists
for an appropriate choice of the generalized sequential averaging method; (ii)
every sequence of weighted ergodic means for the shift operator $x\rightarrow
x\circ\varphi$, $x\in C(\Omega)$, contains a pointwise convergent subsequence.
We also discuss the relationship between the statistical properties of
$(\Omega,\varphi)$ and the mutual structure of minimal sets and ergodic
measures.
