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

KF: Research partially supported by the National Science Centre (Poland) grant 2014/13/ B/ST1/03153.
RS: Supported by NSFC (11201388), NSFC (11271278), ERC starter grant DLGAPS 279893.
CU: Supported by the ERC, via the Starting Grant ChaParDyn, as well as by the Leverhulme Trust via a Leverhulme Prize and by the Royal Society and the Wolfson Foundation via a Royal Society Wolfson Research Merit Award. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 335989.
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  • 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. 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 $\mathscr{H}(1,1)$ of translation surfaces. For these curves 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, 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. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

    Mathematics Subject Classification: Primary: 37A17; Secondary: 37A40, 37J35, 11K38.


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  • Figure 1.  The table $\mathscr{D}_{\lambda_0}$ and two types of invariant sets for the billiard flow

    Figure 2.  Eaton lens and a parallel family of light rays

    Figure 3.  The system of lenses $L(\Lambda,R)$

    Figure 4.  A translation surface $(M, \omega)$ in the space $\mathscr{M}^{dc}$ of double covers of tori.

    Figure 5.  The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $\lambda_0 <\lambda< b$

    Figure 6.  The billiard flow on $\mathscr{S}_{\lambda}$ and the corresponding table $\mathscr{P}_\lambda$ for $b<\lambda< a$

    Figure 7.  The surface $({M}_\lambda, \omega_\lambda)$

    Figure 8.  The system of lenses $F(\Lambda,R,\theta)$

    Figure 9.  The surface $M(\Lambda,R)$

    Figure 10.  The triangle $\triangle ABC$ and the line $x = 2sy+s^2$

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