Extreme events for horocycle flows

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 Poincaré 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.

arXiv comment21 pages, 4 figures

References in corpus (11)

• The asymptotic distribution of Frobenius numbers
• Shrinking targets problems for flows on homogeneous spaces
• The distribution of gaps for saddle connections on the octagon
• Entry and return times for semi-flows
• Universal hitting time statistics for integrable flows
• Logarithm laws for unipotent flows on hyperbolic manifolds
• A Fréchet law and an Erdös-Philipp law for maximal cuspidal windings
• Slope Gap Distribution of Saddle Connections on the 2n-gon
• Slope Gap Distributions of Veech Surfaces
• The Minimal Denominator Function and Geometric Generalizations
• The log moments of smallest denominators