Localizing common fixed points of commuting diffeomorphisms of the plane

Details Localizing common fixed points of commuting diffeomorphisms of the plane Localizing common fixed points of commuting diffeomorphisms of the plane

2010-10-13

arxiv.org/abs/1010.2775v2

Mathematics

Dynamical Systems

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

Scientific paper

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 $G$.

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