The generic isometry and measure preserving homeomorphism are conjugate to their powers

TL;DR

This paper proves that the generic measure preserving homeomorphism and isometry are conjugate to all their powers, allowing extensions to actions of the rationals and adèles, and simplifies proofs of conjugacy class meagreness.

Contribution

It demonstrates that the generic measure preserving homeomorphism and isometries are conjugate to their powers and extends these to actions of the rationals and adèles, providing new structural insights.

Findings

Generic measure preserving homeomorphisms are conjugate to all their powers.

Generic isometries of the rational Urysohn space are conjugate to their powers.

Conjugacy classes are meager in automorphism groups of standard probability spaces and Urysohn space.

Abstract

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of $(\mathbb Q,+)$ by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring $\mathfrak A$ of finite ad\`eles. Similarly, S. Solecki has proved that there is a comeagre set of mutually conjugate isometries of the rational Urysohn metric space. We prove that these are all conjugate with their powers and therefore also embed into $\mathbb Q$-actions. In fact, we extend these actions to actions of $\mathfrak A$ as in the case of measure preserving homeomorphisms. We also consider a notion of topological similarity in Polish groups and use this to give simplified proofs of the meagreness of conjugacy classes in the automorphism group of the standard probability space and in the isometry group of the Urysohn metric space.