Closed subgroups generated by generic measure automorphisms

TL;DR

This paper demonstrates that for a generic measure-preserving transformation, the generated closed subgroup has a rich structure, being a continuous homomorphic image of a subspace of $L_0$ and containing dense unions of finite-dimensional tori.

Contribution

It establishes a novel structural description of the closed groups generated by generic measure automorphisms, linking them to linear subspaces of $L_0$ and torus sequences.

Findings

Closed group is a continuous homomorphic image of a subspace of $L_0$

Contains an increasing sequence of finite-dimensional tori

Union of tori is dense in the group

Abstract

We prove that for a generic measure preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda,{\mathbb R})$, where $\lambda$ is Lebesgue measure, and that the closed group generated by $T$ contains an increasing sequence of finite dimensional toruses whose union is dense.