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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)
##### Keywords:
Teichmüller space; random walk; mapping class group
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##### References:
 [1] B. Bárány, M. Pollicott, and K. Simon, Stationary measures for projective transformations: The Blackwell and Furstenberg measures , J. Stat. Phys. 148 (2012), 393-421. · Zbl 1284.37007 [2] C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials , Ergodic Theory Dynam. Systems 29 (2009), 767-816. · Zbl 1195.37030 [3] B. H. Bowditch, Tight geodesics in the curve complex , Invent. Math. 171 (2008), 281-300. · Zbl 1185.57011 [4] A. I. Bufetov, Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials , J. Amer. Math. Soc. 19 (2006), 579-623. · Zbl 1100.37002 [5] Y. Derriennic, Marche aléatoire sur le groupe libre frontière de Martin , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 261-276. · Zbl 0364.60117 [6] B. Deroin, V. Kleptsyn, and A. Navas, On the question of ergodicity of minimal group actions on the circle , Mosc. Math. J. 9 (2009), 263-303. · Zbl 1193.37034 [7] N. Dunfield and D. Thurston, A random tunnel number one $$3$$-manifold does not fiber over the circle , Geom. Topol. 10 (2006), 2431-2499. · Zbl 1139.57018 [8] B. Farb and D. Margalit, A Primer on Mapping Class Groups , Princeton Math. Ser. 49 , Princeton Univ. Press, Princeton, 2011. · Zbl 1245.57002 [9] B. Farb and H. Masur, Superrigidity and mapping class groups , Topology 37 (1998), 1169-1176. · Zbl 0946.57018 [10] V. S. Gadre, Dynamics of non-classical interval exchanges , Ergodic Theory Dynam. Systems 32 (2012), 1930-1971. · Zbl 1275.37021 [11] Y. Guivarc’h and Y. Le Jan, Asymptotic winding of the geodesic flow on modular surfaces and continued fractions , Ann. Sci. Éc. Norm. Supér. (4) 26 , (1993), 23-50. · Zbl 0784.60076 [12] U. Hamenstädt, “Train tracks and the Gromov boundary of the complex of curves” in Spaces of Kleinian Groups , London Math. Soc. Lecture Note Ser. 329 , Cambridge Univ. Press, Cambridge, 2006, 187-207. · Zbl 1117.30036 [13] V. A. Kaimanovich and V. Le Prince, Matrix random products with singular harmonic measure , Geom. Dedicata 150 (2011) 257-279. · Zbl 1226.60105 [14] V. A. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group , Invent. Math. 125 (1996), 221-264. · Zbl 0864.57014 [15] S. P. Kerckhoff, Simplicial systems for interval exchange maps and measured foliations , Ergodic Theory Dynam. Systems 5 (1985), 257-271. · Zbl 0597.58024 [16] E. Klarreich, The boundary at infinity of the curve complex , preprint, 1999, . · Zbl 1011.30035 [17] F. Ledrappier, “Applications of dynamics to compact manifolds of negative curvature” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) , Birkhäuser, Basel, 1995, 1195-1202. · Zbl 0841.53037 [18] R. Lyons, Equivalence of boundary measures on covering trees of finite graphs , Ergodic Theory Dynam. Systems 14 (1994), 575-597. · Zbl 0821.58008 [19] J. Maher, Linear progress in the complex of curves , Trans. Amer. Math. Soc. 362 (2010), no. 6, 2963-2991. · Zbl 1232.37023 [20] J. Maher, Random Heegard splittings , J. Topol. 3 (2010), 997-1025. · Zbl 1207.37027 [21] J. Maher, Random walks on the mapping class group , Duke Math. J. 156 (2011), 429-468. · Zbl 1213.37072 [22] J. Maher, Harmonic measure for subsurface projections , personal communication, 2009. [23] H. A. Masur, Interval exchange transformations and measured foliations , Ann. of Math. (2) 115 (1982), 169-200. · Zbl 0497.28012 [24] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, I: Hyperbolicity , Invent. Math. 138 (1999), 103-149. · Zbl 0941.32012 [25] H. A. Masur and Y. N. Minsky, Geometry of the complex of curves, II: Hierarchical structure , Geom. Funct. Anal. 10 (2000), 902-974. · Zbl 0972.32011 [26] H. A. Masur, L. Mosher, and S. Schleimer, On train-track splitting sequences , Duke Math. J. 161 (2012), 1613-1656. · Zbl 1275.57029 [27] C. McMullen, Barycentric subdivision and aspect ratios of triangles , personal communication, 2010. [28] R. C. Penner and J. L. Harer, Combinatorics of Train Tracks Ann. of Math. Stud. 125 , Princeton Univ. Press, Princeton, 1992. · Zbl 0765.57001 [29] Y. Peres and B. Solomyak, Absolute continuity of Bernoulli convolutions, a simple proof , Math. Res. Lett. 3 (1996), 231-239. · Zbl 0867.28001 [30] B. Solomyak, On the random series $$\sum\pm\lambda^{n}$$ (an Erdös problem) , Ann. of Math. (2) 142 (1995), 611-625. · Zbl 0837.28007 [31] D. W. Stroock, Probability Theory, an Analytic View , Cambridge Univ. Press, Cambridge, 1993. · Zbl 0925.60004 [32] W. Woess, Random Walks on Infinite Graphs and Groups , Cambridge Tracts Math. 138 , Cambridge Univ. Press, Cambridge, 2000. · Zbl 0951.60002 [33] J.-C. Yoccoz, “Continued fraction algorithms for interval exchange maps: An introduction” in Frontiers in Number Theory, Physics, and Geometry. I , Springer, Berlin, 2006, 401-435. · Zbl 1127.28011
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