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A Fréchet law and an Erdős–Philipp law for maximal cuspidal windings

Published online by Cambridge University Press:  18 July 2012

JOHANNES JAERISCH ,
MARC KESSEBÖHMER  and
BERND O. STRATMANN
JOHANNES JAERISCH
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan (email: jaerisch@cr.math.sci.osaka-u.ac.jp)
MARC KESSEBÖHMER
Affiliation:
Fachbereich 3—Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
BERND O. STRATMANN
Affiliation:
Fachbereich 3—Mathematik und Informatik, Universität Bremen, Bibliothekstraße 1, 28359 Bremen, Germany (email: mhk@math.uni-bremen.de, bos@math.uni-bremen.de)
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Abstract

In this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying extreme value theory. Subsequently, we show that this law gives rise to an Erdős–Philipp law and to various generalized Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdős.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1008 - 1028
Copyright
Copyright © 2012 Cambridge University Press 

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