A Fr\'{e}chet law and an Erd\"os-Philipp law for maximal cuspidal windings

TL;DR

This paper proves a Fréchet law for maximal cuspidal windings in geodesic flows on certain Riemannian surfaces, extending previous results and deriving Erd"os-Philipp laws and Khintchine-type results using Extreme Value Theory.

Contribution

It establishes a new Fréchet law for cuspidal windings in a broad class of Fuchsian groups, extending prior work and connecting it to Erd"os-Philipp laws and Khintchine-type results.

Findings

Proves a Fréchet law for maximal cuspidal windings.

Derives Erd"os-Philipp law and Khintchine-type results from this law.

Extends previous results for Kleinian groups and continued fractions.

Abstract

In this paper we establish a Fr\'{e}chet 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 Erd\"os-Philipp law and to various generalised 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 Erd\"os.