Word length statistics and Lyapunov exponents for Fuchsian groups with cusps

Gadre, V. , Maher, J. and Tiozzo, G. (2015) Word length statistics and Lyapunov exponents for Fuchsian groups with cusps. New York Journal of Mathematics, 21, pp. 511-531.

Abstract

Given a Fuchsian group with at least one cusp, Deroin, Kleptsyn and Navas define a Lyapunov expansion exponent for a point on the boundary, and ask if it vanishes for almost all points with respect to Lebesgue measure. We give an affirmative answer to this question, by considering the behavior of word metric along typical geodesic rays and their excursions into cusps. We also consider the behavior of word length along rays chosen according to harmonic measure on the boundary, arising from random walks with finite first moment. We show that the excursions have different behavior in the Lebesgue measure and harmonic measure cases, which implies that these two measures are mutually singular.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gadre, Dr Vaibhav
• Authors: Gadre, V., Maher, J., and Tiozzo, G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: New York Journal of Mathematics
• Publisher: University of Albany
• ISSN: 1076-9803
• ISSN (Online): 1076-9803
• Copyright Holders: Copyright © 2015 The Authors
• First Published: First published in New York Journal of Mathematics 21:511-531
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.