2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143064Li-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.16 pages, 12 figuresGeometric TopologyDynamical Systems37E30, 57M25 (Primary) 57M50 (Secondary)text