Extreme events for horocycle flows

TL;DR

This work proves extreme value laws for cusp excursions of horocycle flows on finite-area hyperbolic surfaces by reducing the problem to hitting times of shrinking Poincaré sections that scale simply under the geodesic flow. The authors connect these hitting-time laws to hyperbolic lattice-point counting, then to Euclidean point-set statistics, yielding explicit limit densities. In the modular case, the limit density coincides with Hall’s gap distribution for the Farey sequence, and the results extend to general surfaces with multiple cusps, including multi-cusp joint laws and intensity formulas. The paper thereby links cusp-excursion extremes to Farey-gap statistics and saddle-connection gaps on flat surfaces, with potential extensions to infinite-volume settings and moduli spaces of flat structures.

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.

Figures (4)

• Figure 1: The limit density for extreme cusp excursions for the modular surface, which is derived in Section \ref{['secExtreme']}. This is the same distribution, up to scaling and reflection, as for the logs of smallest denominators smalld2.
• Figure 2: The unstable horocycle flow $h^+$ on ${\mathcal{X}}$ can be represented as a flow along horocycles in the complex upper half plane ${\mathbb H}$, moving in counterclockwise direction with unit tangent vectors pointing outwards. The orbit of $x_0$ has a locally maximal excursion into the cusp at time $0 \leq s \leq T$, and hyperbolic distance from ${\mathcal{C}}_\kappa$ greater than $R\geq 0$. The dashed region represents a fundamental domain in ${\mathbb H}$ of the fundamental group $\Gamma$ of the surface ${\mathcal{Y}}$.
• Figure 3: An exercise in hyperbolic geometry: The figure shows a horocycle through the points $0$, $\ell+\mathrm{i} a$ and $\mathrm{i} b$, with unit tangent vectors pointing outwards, the unstable horocycle flow $h^+$ moves points in counter-clockwise motion. The objective is to calculate how long it takes to move with unit speed along the horocycle from the point $\ell+\mathrm{i} a$ to $\mathrm{i} b$, with $l\geq 0$. That is, we wish to compute the length of the horocyclic segment from $\ell+\mathrm{i} a$ to $\mathrm{i} b$. The Möbius transformation $z\mapsto -1/z$ maps this horocycle to the horocycle ${\mathcal{C}}'=\{x+\mathrm{i} b^{-1} : x\in{\mathbb R}\}$. The point $\mathrm{i} b$ is mapped to $\mathrm{i} b^{-1}$ and the point $\ell+\mathrm{i} a$ to $\frac{-\ell+\mathrm{i} a}{\ell^2+a^2}$. Since this point has to lie on ${\mathcal{C}}'$, we have $b^{-1}= \frac{a}{\ell^2+a^2}$, and so $\ell= (ab-a^2)^{1/2}$. Since Möbius transformations act by isometries, the length of the horocyclic segment from $\ell+\mathrm{i} a$ to $\mathrm{i} b$ is the same as the length of the horocyclic segment from $\mathrm{i} b^{-1}$ to $\frac{-\ell+\mathrm{i} a}{\ell^2+a^2}= -\frac{\ell}{ab}+\mathrm{i} b^{-1}$. The latter evaluates to $\frac{\ell}{a}=(\frac{b}{a}-1)^{1/2}$. This relation yields formula \ref{['sejour']} for half of the séjour time $\delta_j$ now follows by setting $a=\mathrm{e}^R$ and $b=\mathrm{e}^{R+t_j}$.
• Figure 4: Schematic illustration for the triangle inequality leading to \ref{['tri']}. The relevant triangle is highlighted in yellow.

Theorems & Definitions (40)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Theorem 2
• proof
• Corollary 2.1
• proof
• Theorem 3
• proof
• Proposition 3.1