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.