Enveloping semigroups and quasi-discrete spectrum

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$-property.

Ergodic properties of discrete dynamical systems and enveloping semigroups

For a continuous semicascade on a metrizable compact set ${\rm\Omega}$, we consider the weak$^{\ast }$ convergence of generalized operator ergodic means in $\text{End}\,C^{\ast }({\rm\Omega})$. We discuss conditions under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on ${\rm\Omega}$, 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 associated algebraic-topological objects: the Ellis semigroup $E$, the Köhler operator semigroup ${\rm\Gamma}\subset \text{End}\,C^{\ast }({\rm\Omega})$, and the semigroup $G=\overline{\text{co}}\,{\rm\Gamma}$. The main results are stated for semicascades with metrizable $E$ and for tame semicascades.

Enveloping semigroups of unipotent affine transformations of the torus

We provide a description of the enveloping semigroup of the affine unipotent transformation T:X→X of the form T(x)=Ax+α, where A is a lower triangular unipotent matrix, α is a constant vector, and X is a finite-dimensional torus. In particular, we show that in this case the enveloping semigroup is a nilpotent group whose nilpotency class is at most the dimension of the underlying torus.