Representations of Artin’s braid groups and linking numbers of periodic orbits

Abstract

Let P be a periodic orbit of period n≥3 of an orientation-preserving homeomorphism f of the 2-disc. Let q be the least integer greater than or equal to n/2−1. Then f admits a periodic orbit Q of period less than or equal to q such that the linking number of P about Q is non-zero. This answers a question of Franks in the affirmative in the case that P has small period. We also derive a result regarding matrix representations of Artin’s braid groups. Finally a lower bound for the topological entropy of a braid in terms of the trace of its Burau matrix is found.