  Li-York theorem tells us that a period 3 orbit for a continuous map of the
interval into itself implies the existence of a periodic orbit of every period.
This paper concerns an analogue of the theorem for homeomorphisms of the
2-dimensional disk. In this case a periodic orbit is specified by a braid type
and on the set of all braid types Boyland's dynamical partial order can be
defined. We describe the partial order on a family of braids and show that a
period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map
which is a factor possessing complicated chaotic dynamics.
