  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 $G$.

Dynamical Systems