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

Christian Rosendal. "The generic isometry and measure preserving homeomorphism are conjugate to their powers." Fundamenta Mathematicae 205.1 (2009): 1-27. <http://eudml.org/doc/282842>.

@article{ChristianRosendal2009,
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 (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. },
author = {Christian Rosendal},
journal = {Fundamenta Mathematicae},
keywords = {comeager conjugacy class; measure-preserving homeomorphism; Cantor space; isometry group; rational Urysohn metric space; Rokhlin's Lemma; topological similarity; finite adèles},
language = {eng},
number = {1},
pages = {1-27},
title = {The generic isometry and measure preserving homeomorphism are conjugate to their powers},
url = {http://eudml.org/doc/282842},
volume = {205},
year = {2009},
}

TY - JOUR
AU - Christian Rosendal
TI - The generic isometry and measure preserving homeomorphism are conjugate to their powers
JO - Fundamenta Mathematicae
PY - 2009
VL - 205
IS - 1
SP - 1
EP - 27
AB - 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.
LA - eng
KW - comeager conjugacy class; measure-preserving homeomorphism; Cantor space; isometry group; rational Urysohn metric space; Rokhlin's Lemma; topological similarity; finite adèles
UR - http://eudml.org/doc/282842
ER -