Localizing common fixed points of commuting diffeomorphisms of the plane

Abstract

We prove that if $G\subset {\text{Diff}}^1({\mathbb{R}}^2)$ is an Abelian subgroup generated by a family of commuting diffeomorphisms of the plane, all of which are $C^1$-close to the identity in the strong $C^1$-topology, and if there exist a point $p\in {\mathbb{R}}^2$ whose orbit is bounded under the action of $G$, then the elements of $G$ have a common fixed point in the convex hull of $\overline{{\mathcal{O}}_p(G)}$. Here, $\overline{{\mathcal{O}}_p(G)}$ denotes the topological closure of the orbit of $p$ by $G$.

arXiv comment19 pages, 8 figures. Final version. To appear in the Bulletin of the Brazilian Mathematical Society