Fixed points of pointwise almost periodic homeomorphisms on the two-sphere

Abstract:

A homeomorphism $f$ of the two-sphere ${S^2}$ onto itself is defined to be pointwise almost periodic (p.a.p.) if the collection of orbit closures forms a decomposition of ${S^2}$. It is shown that if $f:{S^2} \to {S^2}$ is p.a.p. and orientation-reversing then the set of fixed points of $f$ is either empty or a simple closed curve; if $f:{S^2} \to {S^2}$ is p.a.p. orientation-preserving and has a finite number of fixed points, then $f$ is shown to have exactly two fixed points.