# Generic properties of extensions

## TL;DR

This paper demonstrates that in a broad class of measure-preserving systems, weakly mixing extensions are typical, and such generic extensions lack intermediate nilfactors, extending classical results on mixing properties.

## Contribution

It establishes the genericity of weakly but not strongly mixing extensions on fixed product spaces with non-atomic measures, generalizing classical results.

## Key findings

- Weakly but not strongly mixing extensions are generic.
- Generic extensions do not possess intermediate nilfactors.
- Extends classical results by Halmos, Rokhlin, and Furstenberg.

## Abstract

Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nilfactor.