Homeomorphism of S^1 and Factorization

TL;DR

This paper investigates the structure of certain homeomorphism groups of the circle, showing that specific families of transformations are free of relations and exploring their factorization and smoothness properties.

Contribution

It establishes the freeness of families of circle homeomorphisms formed by conjugation with $z o z^n$, and analyzes their role as basic building blocks for more complex groups.

Findings

Families are free of relations, determining the structure of finite type homeomorphism groups.

Connections between smoothness of homeomorphisms and decay of parameters.

Insights into factorization problems for circle homeomorphisms.

Abstract

For each $n > 0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations `conjugated by $z \to z^n$'. We show that these families are free of relations, which determines the structure of `the group of homeomorphisms of finite type'. We also discuss a number of questions regarding factorization for more robust groups of homeomorphisms of the circle in terms of these basic building blocks, and the correspondence between smoothness properties of the homeomorphisms and decay properties of the parameters.