Thurston Elements in the Mapping Class Group

Thurston gave an explicit construction of pseudo-Anosov elements

in the mapping class group of a compact surface, using Dehn twists in pairs of filling multicurves.  We show that the probability that a random walk of length n on the mapping class group gives an element which has a power arising from Thurston’s construction tends to zero as n tends to infinity.  This is joint work with Jing Tao.