Commuting homeomorphisms of \(S^ 2\)
It was proven by \textit{C. Bonatti} [Ann. Math., II. Ser. 129, 61-69 (1989; Zbl 0689.57019)] that, if \(f,g\in \text{Diff}^ 1_ +(S^ 2)\) commute and are sufficiently \(C^ 1\) close to the identity, then they have a common fixed point. This answered a question of H. Rosenberg concerning existence of closed leaves in a certain class of foliations. In this paper, the author, after observing that commuting diffeomorphisms of \(S^ 2\) do not generally have common fixed points, gives a sufficient condition that they do, which generalizes Bonatti's theorem. If \(f,g \in \text{Diff}^ 1_ +(S^ 2)\) commute, he defines a kind of ``winding number'' \(W(f,g) \in \mathbb{Z}_ 2\) and proves that the vanishing of \(W(f,g)\) implies the existence of a common fixed point for \(f\) and \(g\). In fact, \(W(f,g)\) is defined for commuting \(f,g\in \text{Homeo}_ +(S^ 2)\) and, if \(f\) and \(g\) each have finite fixed point sets, the author proves that \(W(f,g) = 0\) implies that \(f\) and \(g\) have a common fixed point. Moreover, if at least one of \(f\) and \(g\) has more than one fixed point, then they have in common at least two fixed points. These are the main theorems and related results are proven for commuting homeomorphisms of \(\mathbb{R}^ 2\) and of closed orientable surfaces.
scientific article; zbMATH DE number 89835