The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids

TL;DR

This paper explores the partial order of braid types in disk homeomorphisms, showing that a period 3 pseudo-Anosov braid implies chaotic dynamics similar to the Smale-horseshoe map.

Contribution

It introduces a new partial order on braid types and demonstrates its implications for chaotic behavior in disk homeomorphisms.

Findings

Period 3 pseudo-Anosov braid type implies Smale-horseshoe dynamics

Describes the partial order structure on a family of braids

Connects braid type order to chaotic dynamics in disk maps

Abstract

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.