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

TL;DR

This paper proves that harmonic measures arising from finite support distributions on the mapping class group are singular relative to Lebesgue measure on the space of projective measured foliations, highlighting a distinct measure-theoretic property.

Contribution

It establishes the singularity of harmonic measures for finite support distributions on the mapping class group, extending previous results on harmonic measures.

Findings

Harmonic measures for finite support distributions are singular.

Supports of these measures do not coincide with Lebesgue measure.

The result contrasts with measures from more general distributions.

Abstract

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on 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 on PMF.