Distortion element in group of diffeomorphisms of the 2-sphere

TL;DR

This paper characterizes distortion elements in the diffeomorphism group of the 2-sphere, showing they are irrational pseudo-rotations under certain conditions and analyzing their differential eigenvalues.

Contribution

It establishes new properties of distortion elements in the 2-sphere diffeomorphism group, including their classification as irrational pseudo-rotations and eigenvalue uniqueness.

Findings

Distortion elements with recurrent points are irrational pseudo-rotations.

Such elements have a differential with a unique eigenvalue of 1 at fixed points.

Provides new insights into the structure of diffeomorphism groups of the 2-sphere.

Abstract

We prove that every distortion element in the group of diffeomorphisms of the 2-sphere which has some recurrent point that is not fixed is an irrational pseudo-rotation. Moreover we prove that the differential of a distortion element in the group of diffeomorphisms of the 2-sphere having at least three fixed points at a fixed point has a unique eigenvalue which is 1.