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

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 (R)/ASL 2 (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 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 Dragović and Radnović, 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 Frączek 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.

Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space

“…, e −t ). In a joint work with Fraczek and Ulcigrai [10], we prove pointwise equidistribution for certain curves which are parameterized by a horospherical subgroup. Theorem 1.1 is deduced from Theorem 1.2 and the asymptotic equidistribution of measures proved in [29].…”

Self Affine Delone Sets and Deviation Phenomena

“…For example, the rates dictated by the Lyapunov exponents of the Kontsevich-Zorich cocycle have been shown to control the limiting large-scale geometry of associated systems, some of which come from physical models (see e.g. [DHL14,FSU15]). We consider Theorem 4 to be of this type.…”

On Ergodicity of Foliations on $${\mathbb{Z}^d}$$ Z d -Covers of Half-Translation Surfaces and Some Applications to Periodic Systems of Eaton Lenses

“…More precisely, a light ray in an Eaton lens configuration is called trapped, if the ray never leaves a strip parallel to a line in R 2 . The trapping phenomenon observed in [17] was extended in [16] to the following result: Theorem 1.1. If L( , R) is an admissible configuration then for a.e.…”

“…Among other things, Theorem 2.4 in [19] says, that for all Riemann metrics on the plane that are pull backs of Riemann metrics on a torus with vanishing topological entropy, the geodesics are trapped. Nevertheless, the trapping phenomena obtained in [16][17][18][19] have different flavors. The former is transient whereas the latter is recurrent.…”