Most homeomorphisms with a fixed point have a Cantor set of fixed points

Abstract

We show that, for any n ≠ 2, most orientation preserving homeomorphisms of the sphere S ²ⁿ have a Cantor set of fixed points. In other words, the set of such homeomorphisms that do not have a Cantor set of fixed points is of the first Baire category within the set of all homeomorphisms. Similarly, most orientation reversing homeomorphisms of the sphere S ²ⁿ⁺¹ have a Cantor set of fixed points for any n ≠ 0. More generally, suppose that M is a compact manifold of dimension > 1 and ≠ 4 and ${\mathcal{H}}$ is an open set of homeomorphisms h : M → M such that all elements of ${\mathcal{H}}$ have at least one fixed point. Then we show that most elements of ${\mathcal{H}}$ have a Cantor set of fixed points.