Des points fixes communs pour des diff\'eomorphismes de S^2 qui commutent et pr\'eservent une mesure de probabilit\'e

TL;DR

This paper proves that commuting homeomorphisms of the plane or sphere that preserve a probability measure have common fixed points, with specific results for diffeomorphisms close to the identity.

Contribution

It establishes the existence of common fixed points for commuting measure-preserving homeomorphisms and diffeomorphisms on the sphere and plane, extending fixed point results.

Findings

Commuting measure-preserving homeomorphisms on S^2 have common fixed points.

Certain C^1-diffeomorphisms close to identity have at least two common fixed points.

Results apply to transformations with non-trivial support of the invariant measure.

Abstract

We prove the existence of common fixed points for commuting homeomorphisms of the plane R^2 or the sphere S^2, which preserve a probability measure. For example: some commuting C^1-diffeomorphisms of S^2, which are sufficiently close to the identity and preserve a probability measure whose support is not a single point, have at least two common fixed points.