Relative weak mixing is generic

TL;DR

This paper extends the classical result that weak mixing is generic among measure-preserving transformations to ergodic extensions of a fixed transformation and further to actions of countable amenable groups.

Contribution

It generalizes Halmos' theorem to ergodic extensions and countable amenable groups, broadening the scope of weak mixing's genericity.

Findings

Weak mixing is generic among ergodic extensions of a fixed transformation.

The result extends to actions of countable amenable groups.

The paper uses a combination of classical and modern ergodic theory techniques.

Abstract

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergodic extensions of a fixed, but arbitrary, ergodic transformation $T_0$. We then use a result of Connes, Feldman and Weiss to extend this relative theorem to the general (countable) amenable group.