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

Christian Rosendal¹

¹ Department of Mathematics, Statistics, and Computer Science (M//C 249) University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607-7045, U.S.A.

Fundamenta Mathematicae, Tome 205 (2009) no. 1, pp. 1-27

Résumé

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 adèles.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.

Détail

DOI : 10.4064/fm205-1-1

Keywords: known there comeagre set mutually conjugate measure preserving homeomorphisms cantor space equipped coinflipping probability measure haar measure generic measure preserving homeomorphism moreover conjugate its powers follows generic measure preserving homeomorphism extends action mathbb measure preserving homeomorphisms action locally compact ring mathfrak finite les similarly solecki has proved there comeagre set mutually conjugate isometries rational urysohn metric space prove these conjugate their powers therefore embed mathbb q actions extend these actions actions mathfrak measure preserving homeomorphisms consider notion topological similarity polish groups simplified proofs meagreness conjugacy classes automorphism group standard probability space isometry group urysohn metric space