We show that vertical light rays in almost every periodic array of Eatonlenses do not leave certain strips of bounded width. The light rays are tracedby leaves of a non-orientable foliation on a singular plane. We study the flowdefined by the induced foliation on the orientation cover of the singularplane. The behavior of that flow and ultimately our claim for the light rays isbased on an analysis of the Teichm\"uller flow and the Kontsevich-Zorichcocycle on the moduli space of two branched, two sheeted torus covers in genustwo.
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