# Maximal pseudometrics and distortion of circle diffeomorphisms

## TL;DR

This paper explores distortion elements in groups of circle diffeomorphisms using maximal metrics, classifies distortion in the $C^1$ case, and constructs examples of undistorted diffeomorphisms with specific regularity properties.

## Contribution

It provides a classification of distortion in $C^1$ circle diffeomorphisms and constructs examples of undistorted analytic diffeomorphisms with only non-hyperbolic fixed points.

## Key findings

- A $C^1$ diffeomorphism is undistorted iff it has a hyperbolic periodic point.
- Existence of analytic circle diffeomorphisms with only non-hyperbolic fixed points that are undistorted.
- The group $ ext{Diff}_+^{1+AC}(S^1)$ is quasi-isometric to a hyperplane in $L^1(I)$.

## Abstract

We initiate a study of distortion elements in the Polish groups $\mbox{Diff}_+^k(\mathbb{S}^1)$ ($1\leq k<\infty$), as well as $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, in terms of maximal metrics on these groups. We classify distortion in the $k=1$ case: a $C^1$ circle diffeomorphism is $C^1$-undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are $C^{1+AC}$-undistorted, and hence $C^k$-undistorted for all $k\geq 2$. In the appendix, we exhibit a maximal metric on $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, and observe that this group is quasi-isometric to a hyperplane of $L^1(I)$.