Fixed Points of abelian actions on $S^2$

Franks, John; Handel, Michael; Parwani, Kamlesh

Fixed Points of abelian actions on $S^2$

Authors

Franks, John

Handel, Michael

Parwani, Kamlesh

Description

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.