Topology of the space of measure-preserving transformations of the circle

TL;DR

This paper proves that the space of degree 2 expanding circle maps preserving Lebesgue measure is arc-connected and homeomorphic to an infinite-dimensional Lie group, revealing unexpected topological properties.

Contribution

It establishes the topological structure of the space of measure-preserving expanding maps on the circle, including its arc-connectedness and Lie group homeomorphism.

Findings

The space is arc-connected.

It is homeomorphic to an infinite-dimensional Lie group.

Its fundamental group is isomorphic to .

Abstract

This paper is dedicated to prove that the space of circle expanding maps of degree 2 preserving Lebesgue measure is an arc-connected space homeomorphic to an infinite-dimensional Lie group whose fundamental group is $\mathbb{Z}$. The techniques involved in the proof are rather unexpected and lead to a formulation of a general conjecture