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Recurrence of quadratic differentials for harmonic measure

Part of: Stochastic processes Ergodic theory Dynamical systems with hyperbolic behavior Special aspects of infinite or finite groups

Published online by Cambridge University Press:  25 June 2019

VAIBHAV GADRE  and
JOSEPH MAHER
VAIBHAV GADRE
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ. e-mail: Vaibhav.Gadre@glasgow.ac.uk
JOSEPH MAHER
Affiliation:
Department of Mathematics, College of Staten Island, CUNY 2800 Victory Boulevard, Staten Island, NY 10314, U.S.A and Department of Mathematics, 4307 Graduate Center, CUNY 365 5th Avenue, New York, NY 10016, U.S.A e-mail: joseph.maher@csi.cuny.edu
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Abstract

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 60G50: Sums of independent random variables; random walks

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 299 - 305
Copyright
© Cambridge Philosophical Society 2019

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References

REFERENCES

Algom–kfir, Y., Kapovich, I. and Pfaff, C.. Stable strata of geodesics in outer space. Int. Math. Res. Not. To appear (2019).CrossRefGoogle Scholar
Gadre, V. and Maher, J., The stratum of random mapping classes. Ergodic Theory Dynam. Syst., to appear (2017).CrossRefGoogle Scholar
Kaimanovich, V. A. and Masur, H., The Poisson boundary of the mapping class group. Invent. Math. 125 (1996), No. 2, 221–264.CrossRefGoogle Scholar
Kapovich, I. and Pfaff, C., A train track directed random walk on Out(Fr). Internat. J. Algebra Comput. 25 (2015), no. 5, 745–798.CrossRefGoogle Scholar
Kapovich, I., Maher, J., Pfaff, C., and Taylor, S. J., Random outer automorphisms of free groups: attracting trees and their singularity structures (2018). Available at arXiv:1805.12382.Google Scholar
Kapovich, I., Maher, J., Pfaff, C. and Taylor, S. J., Random trees in the boundary of Outer space (2019) Available at arXiv:1904.10026.Google Scholar
Maher, J., Random walks on the mapping class group. Duke Math. J. 156 (2011), no. 3, 429–468.CrossRefGoogle Scholar
Maskit, B., Comparison of hyperbolic and extremal lengths. Ann. Acad. Sci. Fenn. Ser. A I Math., 10, (1985), 381–386.CrossRefGoogle Scholar
Masur, H., Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J. 66, (1992), 3, 387–442.Google Scholar
Rafi, K., Hyperbolicity in Teichmüller space. Geom. Topol. 18, (2014), no. 5, 3025–3053.CrossRefGoogle Scholar
Rivin, I., Walks on groups, counting reducible matrices, polynomials and surface and free group automorphisms. Duke Math. J. 142, (2008), no. 2, 353–379.CrossRefGoogle Scholar