Abstract

We relate trimmed sums of twists in cylinders along a typical Teichmüller geodesic to the area Siegel–Veech constant.

Highlights

  • The strong law in [10] relates trimmed sums of excursions of a random geodesic in the thin parts of moduli spaces of quadratic differentials to Siegel–Veech constants. This strong law is a generalisation of the Diamond–Vaaler strong law [4] for continued fraction coefficients: for almost every r ∈ [0, 1] the continued fraction coefficients of r satisfy a1 + a2 + · · · + an − max ak lim jn =

  • By work of Series [16], continued fraction coefficients can be interpreted in terms of hyperbolic geodesic rays on the modular surface X = H/S L(2, Z)

  • In the upper half-space model, consider the hyperbolic geodesic γ converging to r ∈ [0, 1] ⊂ R ∪ ∞ = ∂H

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Summary

Introduction

The strong law in [10] relates trimmed sums of excursions of a random geodesic in the thin parts of moduli spaces of quadratic differentials to Siegel–Veech constants. This strong law is a generalisation of the Diamond–Vaaler strong law [4] for continued fraction coefficients: for almost every r ∈ [0, 1] the continued fraction coefficients of r satisfy a1 + a2 + · · · + an − max ak lim jn =. One proves a continuous time strong law for trimmed sums of excursions of hyperbolic geodesics in cusp neighbourhoods of X = H/. It combines the decay of co-relations with estimates coming from the cusp geometry of X From this point, the strong law generalises to cusp excursions of Teichmüller geodesics in the thin parts of S L(2, R)-orbit closures. These constants are important for their relationship with other dynamical quantities such as the Lyapunov spectrum for the Teichmüller flow [6, Theorem 1]

Quadratic differentials
Orbit closures
Siegel–Veech formula
Thin parts
Area Siegel–Veech constants
Excursions
Twists
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