Distortion in the group of circle homeomorphisms

TL;DR

This paper proves constructively that all irrational rotations are distorted elements within the groups of piecewise-affine and smooth circle homeomorphisms, highlighting their geometric properties.

Contribution

It provides a constructive proof demonstrating that irrational rotations are distorted in both the piecewise-affine and smooth circle homeomorphism groups.

Findings

Irrational rotations are distorted in ${ m PAff}_+({f S}^1)$.

Irrational rotations are distorted in ${ m Diff}^{ abla}( ext{ extbf{R}/ extbf{Z}})$.

Constructive proof technique for distortion.

Abstract

Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.