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In this paper we prove results on Birkhoff and Oseledets genericity alongcertain curves in the space of affine lattices and in moduli spaces oftranslation surfaces. We also prove applications of these results to dynamicalbilliards, mathematical physics and number theory. In the space of affinelattices $ASL_2(\mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost everypoint on a curve with some non-degeneracy assumptions is Birkhoff generic forthe geodesic flow. This implies almost everywhere genericity for some curves inthe 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 arewell-approximated by horocycle arcs and satisfy almost everywhere Birkhoffgenericity) we also prove that almost every point is Oseledets generic for theKontsevitch-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 Dragovicand Radnovic, and prove that for almost every parameter, the billiard flow isuniquely ergodic within the region of phase space in which it is trapped. Wethen consider any periodic array of Eaton retroreflector lenses, placed onvertices of a lattice, and prove that in almost every direction light rays areeach confined to a band of finite width. This generalizes a phenomenon recentlydiscovered by Fraczek and Schmoll which could so far only be proved for randomperiodic configurations. Finally, a result on the gap distribution offractional parts of the sequence of square roots of positive integers, whichextends previous work by Elkies and McMullen, is also obtained.

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