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Recurrence of quadratic differentials for harmonic measure

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

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: [email protected]
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: [email protected]
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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
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

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