On some generic classes of ergodic measure preserving transformations

TL;DR

This paper demonstrates that being a relatively weakly mixing extension is a typical property in the space of measure preserving transformations and explores how certain properties are preserved in generic extensions.

Contribution

It proves that relative weak mixing is a comeager property and shows that properties like entropy, Bernoulli, K, and loosely Bernoulli are generically preserved in extensions.

Findings

Relatively weakly mixing extensions are comeager in the space of transformations.

Generic extensions of an ergodic transformation with property A also have property A.

The paper establishes interrelations among classes of ergodic transformations.

Abstract

We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.