Harmonic measures for distributions with finite support on the mapping class group are singular

Abstract

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution whose support generates a nonelementary subgroup when projected into Teichmüller space converges almost surely to a point in the space $\mathcal{PMF}$ of projective measured foliations on the surface. This defines a harmonic measure on $\mathcal{PMF}$. Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure class on $\mathcal{PMF}$.