Pseudo-Anosov Braid Types of the Disc or Sphere of Low Cardinality Imply all Periods

Abstract

We show that any orientation-preserving homeomorphism of the 2-disc possessing either a 3- or 4-point invariant set X either possesses periodic orbits of all periods or belongs to one of a small number of periodic or reducible isotopy classes relative to X. We prove also that for any homeomorphism of the annulus isotopic to the identity which is pseudo-Anosov relative to a finite invariant set, there exist periodic orbits in the interior whose rotation numbers are those of the boundaries. Finally we show that if f is an orientation-preserving homeomorphism of the 2-sphere possessing an invariant set Y of cardinality 4 or 5 such that the braid type of Y is pseudo-Anosov, then f has periodic orbits of all periods.