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The stratum of random mapping classes. (English) Zbl 1409.37040
Summary: 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. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of I. Kapovich and C. Pfaff [Int. J. Algebra Comput. 25, No. 5, 745–798 (2015; Zbl 1351.20025)].

MSC:
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37H10 Generation, random and stochastic difference and differential equations
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
05C81 Random walks on graphs
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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