The Return Map of the Cross Section of Horizontally Short Lattice Surfaces is Weakly Mixing

TL;DR

This work proves that the return map of the unstable horocycle flow on the space of horizontally short translation surfaces associated to a lattice surface is weakly mixing, generalizing the square-torus result. The authors adapt the Cheung–Quas criterion to the ω-BCZ return map on a carefully parameterized Poincaré section $\mathcal{L}$, and leverage a restricted Siegel–Veech transform to obtain precise point-counting formulas. Central contributions include a detailed height-distribution analysis of holonomy vectors, a reduction of the Cheung–Quas hypotheses to verifiable geometric bounds, and a robust second-moment bound for thin-rectangle counts. Together these results extend dynamical insights into translation surfaces beyond the square torus and provide tools for Siegel–Veech-type statistics on moduli spaces with finitely many cusps.

Abstract

We prove that the return map of the unstable horocycle flow on the space of horizontally short translation surfaces associated to a lattice surface $(X, ω)$ is weakly mixing. This extends a result of Cheung-Quas for the square torus to all lattice surfaces. The proof adapts their criterion for weakly mixing and uses quantitative bounds for Siegel-Veech transforms restricted to the Poincaré section of horizontally short surfaces.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 5.1
• proof
• Proposition 5.2