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

2025/10/17 by Albert Artiles, Artiles, Albert
Mathematics · #37A25 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2510.15450

openalex publication_date 2025/10/17 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28

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.

Citations

• An Effective Slope Gap Distribution for Lattice Surfaces
• Gaps of saddle connection directions for some branched covers of tori
• The Boca-Cobeli-Zaharescu Map Analogue for the Hecke Triangle Groups G_q
• The gap distribution of slopes on the golden L
• A Poincaré section for horocycle flow on the space of lattices
• The distribution of gaps for saddle connection directions
• Siegel measures

Related

• Weak mixing for interval exchange transformations and translation flows
• On Mixing Flows on Finite Area Translation Surfaces
• Weak mixing directions in non-arithmetic Veech surfaces
• Twisted Translation Flows and Effective Weak Mixing
• A Poincaré section for horocycle flow on the space of lattices