On metrizable enveloping semigroups

TL;DR

This paper establishes the equivalence between several dynamical properties of metrizable systems, including the metrizability of the enveloping semigroup, extending recent characterizations of hereditarily almost equicontinuous systems.

Contribution

It proves that the conditions related to hereditarily almost equicontinuous systems are also equivalent to the metrizability of the enveloping semigroup, linking topological and Banach space representations.

Findings

Equivalence of hereditarily almost equicontinuous and non-sensitive conditions.

Metrizability of the enveloping semigroup characterizes these dynamical properties.

Connection to proper representations on Asplund Banach spaces.

Abstract

When a topological group $G$ acts on a compact space $X$, its enveloping semigroup $E(X)$ is the closure of the set of $g$-translations, $g\in G$, in the compact space $X^X$. Assume that $X$ is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) $X$ is hereditarily almost equicontinuous; (2) $X$ is hereditarily non-sensitive; (3) for any compatible metric $d$ on $X$ the metric $d_G(x,y):=\sup\{d(gx,gy): g\in G\}$ defines a separable topology on $X$; (4) the dynamical system $(G,X)$ admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup $E(X)$ is metrizable.