# The structure of random homeomorphisms

## TL;DR

This paper investigates the measure-theoretic structure of conjugacy classes in certain homeomorphism groups, revealing that most classes are Haar null and providing insights into their measure-theoretic size and distribution.

## Contribution

It characterizes non-Haar null conjugacy classes in specific homeomorphism groups and demonstrates that their union is co-Haar null, advancing understanding of typical element behavior.

## Key findings

- Non-Haar null conjugacy classes in Homeo^+([0,1]) are described.
- In Homeo^+(), non-Haar null classes are characterized.
- In a(), all classes except multishifts are Haar null.

## Abstract

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For $\text{Homeo}^+([0,1])$ we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for $\text{Homeo}^+(\mathbb{S}^1)$ we describe the non-Haar null conjugacy classes, and for $\mathcal{U}(\ell^2)$ we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.