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

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.Comment: To appear in Journal of Modern Dynamic

Ergodic properties of the ideal gas model for infinite billiards

In this paper we study ergodic properties of the Poisson suspension (the ideal gas model) of the billiard flow $(b_t)_{t\in\mathbb R}$ on the plane with a $\Lambda$-periodic pattern ($\Lambda\subset\mathbb R^2$ is a lattice) of polygonal scatterers. We prove that if the billiard table is additionally rational then for a.e. direction $\theta\in S^1$ the Poisson suspension of the directional billiard flow $(b^\theta_t)_{t\in\mathbb R}$ is weakly mixing. This gives the weak mixing of the Poisson suspension of $(b_t)_{t\in\mathbb R}$. We also show that for a certain class of such rational billiards (including the periodic version of the classical wind-tree model) the Poisson suspension of $(b^\theta_t)_{t\in\mathbb R}$ is not mixing for a.e. $\theta\in S^1$

Normalisation with respect to pattern

The article presents a new normalisation method of diagnostic variables - normalisation with respect to the pattern. The normalisation preserves some important descriptive characteristics of variables: skewness, kurtosis and the Pearson correlation coeffcients. It is particularly useful in dynamical analysis, when we work with the whole population of objects not a sample, for example in regional studies. After proposed transformation variables are comparable not only between themselves but also across time. Then we can use them, for example, to construct composite variables