Dynamics of braids

Abstract

In this thesis we study the dynamics of braids using the folding decomposition of train track maps and the entropy. In Chapter 1 we explain the preliminaries about braids, the Nielsen-Thurston classification of surface homeomorphisms, and the train track algorithm. In Chapter 2 we show that the entropy of a braid can be computed from the spectral radius of the Burau matrix of another braid after reviewing the topological entropy of braid and the Burau representation. In Chapter 3 we introduce the folding decomposition of train track maps and show that for each braid index there are finitely many folding automata which generate all the conjugacy classes of pseudo-Anosov braids. In Chapter 4 we use the folding automata to find the braid with the minimal non-trivial entropy for braid index less than 6. In Chapter 5 we estimate the minimal entropy of braids with general braid index and show that the entropy of a pseudo-Anosov pure braid is greater than or equal to $\log 3$.