A self-homeomorphism f of the
2-sphere S2 is weakly almost periodic (w.a.p.) if the collection of orbit closures forms
a continuous decomposition of S2. It is shown that if f is orientation-preserving,
w.a. p. and nonperiodic, then f has exactly two fixed points, and every
nondegenerate orbit closure is an homology l-sphere. There is an example with an
orbit closure which is an homology l-sphere but not a real l-sphere. If f is
orientation-reversing, w.a. p. and has a fixed point, then f is shown to be periodic.
The orbit structure of orientation-reversing, w.a. p., nonperiodic homeomorphisms
on S2 is studied.
