Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

摘要

Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn andNavas define a Lyapunov expansion exponent for a point on theboundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give anaffirmative answer to this question, by considering the behavior ofthe word metric along typical geodesic rays and their excursions intocusps. We also consider the behavior of the word metric along rays chosenaccording to harmonic measure on the boundary, arising from randomwalks with finite first moment. We show that the excursions havedifferent behavior in the Lebesgue measure and harmonic measurecases, which implies that these two measures are mutually singular.