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

• Autorzy: Christian Rosendal
• Języki publikacji: EN

Abstrakty

EN

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 (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 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 ℚ-actions. In fact, we extend these actions to actions of 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.

Kategorie tematyczne

• 37A05: Measure-preserving transformations
• 03E15: Descriptive set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 205
• Numer: 1
• Strony: 1-27

Daty

wydano

2009

Twórcy

autor

Christian Rosendal

• 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.

Identyfikatory

DOI

10.4064/fm205-1-1