Localizing common fixed points of commuting diffeomorphisms of the plane

TL;DR

This paper proves that for a certain class of commuting plane diffeomorphisms close to the identity, the existence of a bounded orbit guarantees a common fixed point within the convex hull of that orbit.

Contribution

It establishes a fixed point theorem for Abelian subgroups of diffeomorphisms near the identity, linking bounded orbits to common fixed points in the plane.

Findings

Existence of a common fixed point under specified conditions.

Fixed point lies in the convex hull of the orbit closure.

Applicable to Abelian groups generated by commuting diffeomorphisms.

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 $\bar{\mathcal{O}_{p}(G)}$. Here, $\bar{\mathcal{O}_{p}(G)}$ denotes the topological closure of the orbit of $p$ by $G$.