Extreme events for horocycle flows

TL;DR

This paper establishes extreme value laws for cusp excursions in horocycle flows on negatively curved surfaces, linking hitting time distributions to Farey sequence gap distributions, extending previous modular surface results.

Contribution

It introduces a new approach using simple scaling properties of Poincaré sections to derive extreme value laws for horocycle flows on general surfaces.

Findings

Extreme value laws for cusp excursions are proven.

Limit laws relate to Hall's formula for Farey sequence gaps.

Extension of previous modular surface results to broader surfaces.

Abstract

We prove extreme value laws for cusp excursions of the horocycle flow in the case of surfaces of constant negative curvature. The key idea of our approach is to study the hitting time distribution for shrinking Poincar\'e sections that have a particularly simple scaling property under the action of the geodesic flow. This extends the extreme value law of Kirsebom and Mallahi-Karai [arXiv:2209.07283] for cusp excursions for the modular surface. Here we show that the limit law can be expressed in terms of Hall's formula for the gap distribution of the Farey sequence.