Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

TL;DR

This paper proves genericity results for curves in affine lattices and translation surfaces, with applications to billiards, optical systems, and number theory, demonstrating almost everywhere ergodic and distributional properties.

Contribution

It establishes Birkhoff and Oseledets genericity along curves in moduli spaces, extending previous results and applying them to billiards, Eaton lenses, and gap distributions.

Findings

Almost every point on certain curves is Birkhoff generic for geodesic flow.

Almost every point on these curves is Oseledets generic for the Kontsevich-Zorich cocycle.

In almost every direction, light rays in Eaton lenses are confined to finite-width bands.

Abstract

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathcal{H}(1,1)$ of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragovic and Radnovic, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Fraczek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.