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Harmonic measures for distributions with finite support on the mapping class group are singular. (English) Zbl 1285.30025
Summary: V. A. Kaimanovich and H. Masur [Invent. Math. 125, No. 2, 221–264 (1996; Zbl 0864.57014)] 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 \(\mathcal{PMF}\).

MSC:
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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