Veech surfaces associated with rational billiards

Deviation of ergodic averages for rational polygonal billiards

We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.

Arithmetic Veech sublattices of $\SL(2,\Z)$

We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.

Billiards in right triangles and orbit closures in genus zero strata

The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.

Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb{C}P^1$

We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered. An appendix with an ergodic theorem proved by Jon Chaika …

From rational billiards to dynamics on moduli spaces

This short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces, and in particular the recent breakthrough result of Eskin, Mirzakhani, and Mohammadi. We also discuss the context and applications of this result, and connections to other areas of mathematics such as algebraic geometry, Teichmuller theory, and ergodic theory on homogeneous spaces.

弱混合多边形台球

Weakly Mixing Polygonal Billiards

本文证明了存在非有理多边形的剩余集，使得台球流关于Liouville测度弱混合，其核心方法是通过Baire范畴论证及平移曲面方向对流的遍历性与特征值性质。