On weak rigidity and weakly mixing enveloping semigroups

TL;DR

This paper investigates whether nontrivial dynamical systems can have enveloping semigroups that are topologically weakly mixing, establishing necessary conditions and providing examples related to horocycle flows.

Contribution

It introduces necessary conditions for weakly mixing enveloping semigroups and demonstrates that the time one map of a horocycle flow has a weakly mixing enveloping semigroup.

Findings

Necessary conditions for weakly mixing enveloping semigroups

The enveloping semigroup of a horocycle flow's time one map is weakly mixing

Application of Ratner's theory to dynamical systems

Abstract

The question we deal with here, which was presented to us by Joe Auslander and Anima Nagar, is whether there is a nontrivial cascade (X,T) whose enveloping semigroup, as a dynamical system, is topologically weakly mixing (WM). After an introductory section recalling some definitions and classic results, we establish some necessary conditions for this to happen, and in the final section we show, using Ratner's theory, that the enveloping semigroup of the `time one map' of a classical horocycle flow is weakly mixing.