Fixed Points of abelian actions on $S^2$

TL;DR

This paper proves that finitely generated abelian groups of orientation-preserving $C^1$ diffeomorphisms on $R^2$ with invariant compact sets have a common fixed point, and similar results hold on $S^2$ with a subgroup of index at most two.

Contribution

It establishes fixed point results for abelian groups of $C^1$ diffeomorphisms on $R^2$ and $S^2$, extending classical fixed point theorems to these settings.

Findings

Finitely generated abelian groups on $R^2$ have a common fixed point if they leave a compact set invariant.

Any abelian subgroup of orientation-preserving $C^1$ diffeomorphisms on $S^2$ has a subgroup of index at most two with a common fixed point.

The results extend fixed point theorems to broader classes of smooth group actions on surfaces.

Abstract

We prove that if $F$ is a finitely generated abelian group of orientation preserving $C^1$ diffeomorphisms of $R^2$ which leaves invariant a compact set then there is a common fixed point for all elements of $F.$ We also show that if $F$ is any abelian subgroup of orientation preserving $C^1$ diffeomorphisms of $S^2$ then there is a common fixed point for all elements of a subgroup of $F$ with index at most two.