Billiards in L-Shaped Tables with Barriers

Abstract

We compute the volumes of the eigenform loci in the moduli space of genus-two Abelian differentials. From this, we obtain asymptotic formulas for counting closed billiards paths in certain L-shaped polygons with barriers.

Siegel–Veech Constants for Strata of Moduli Spaces of Quadratic Differentials

“…Application of Siegel-Veech constants. One of the principal reasons why the Siegel-Veech constants are more and more intensively studied during the last years [1,3,4,6,14,34] is the relation between them and the Lyapunov exponents of the Hodge bundle along the Teichmüller flow: the key formula of [14] expresses the sum of the positive Lyapunov exponents for any stratum Q(α) as a sum of a very explicit rational function in α and the Siegel-Veech constant c area (Q(α)). The Lyapunov exponents are closely related to the deviation spectrum of measured foliations on individual flat surfaces [21,22,36,37], which opens applications to billiards in polygons, interval exchanges, etc.…”

GL+(2, ℝ)–orbits in Prym eigenform loci

“…Along the way, we give a partial proof that the compactification of PΩE D (2, 2) odd in PΩM 3 is an algebraic variety. In the case of genus two, this result was proved by McMullen [McM05b,McM06] and Bainbridge [Ba07,Ba10].…”

Translation surfaces with no convex presentation

“…The following result is proved in [Mc3,B] Theorem 22 (McMullen, Bainbridge). Suppose θ is a completely periodic direction on an eigenform surface of discriminant D in H(1, 1), such that the corresponding cylinder decomposition has three cylinders, or two cylinders with one simple cylinders.…”