Intersection Points of Closed Geodesics on Hyperbolic Surfaces of Finite Area

TL;DR

The paper proves that the intersection points of all closed geodesics of length at most T on a complete finite-area hyperbolic surface X become uniformly distributed with respect to the area measure as T grows. It develops a geodesic-current framework, introducing the intersection measure I(C1, C2) and the Liouville current L_X to lift equidistribution from closed geodesics in the unit tangent bundle to their intersection points on X, while addressing cusp phenomena with a cusp-excursion analysis that yields tightness. A key technical achievement is a bound on excursions E_n(γ_T) that prevents mass escape to cusps, enabling the main convergence I(γ_T, γ_T)/|I(γ_T, γ_T)| → α/area(X) and its corollaries. The results further imply equidistribution of intersection directions and, for any finite arc η, convergence of the induced intersection measure to η's length measure, thereby providing a robust, geometric description of intersection statistics on finite-area hyperbolic surfaces.

Abstract

Let X be a complete hyperbolic surface of finite area. We establish that the intersection points of closed geodesics with length <T are equidistributed on X as T goes to infinity.

Figures (6)

• Figure 1: An $H-$Box
• Figure 2: An example that shows the intersection number is not continuous
• Figure 3: A homotopic modification of $\alpha$ and $\beta$. The curves $\alpha'$ and $\beta'$ coincide with geodesics orthogonal to $\gamma$ within the regions $R_2$ and $R_1$, respectively.
• Figure 4: The preimage of an $n-$excursion in the upper half-plane
• Figure 5: How the map $\psi$ modifies a curve by removing an $n-$excursion

Theorems & Definitions (34)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Corollary 2.4
• proof