Simple Braids Tend toward Positive Entropy

A simple braid is a positive braid that can be drawn so that any two strands cross at most once. We prove that as $n\to \mathrm{\infty }$, the proportion of simple braids on $n$ strands that have positive topological entropy tends toward $100\mathrm{\%}$. Notably, such braids are either pseudo-Anosov or reducible with a pseudo-Anosov component. Our proof involves a method of reduction from simple braids to non-simple $3$-strand braids that may be of independent interest.