On ergodicity in periodic systems of Eaton lenses

We consider periodic Eaton lens patterns in the plane. An Eaton lens is a circular lens on the plane which acts as a perfect retroreflector, i.e. so that each ray of light after passing through the Eaton lens is directed back toward its source. The aim of the talk is to present some relations between ergodic properties of the geodesic flow on the plane equiped periodic systems of lenses and vanishing of Lyapunov exponents of quadratic differentials on the torus.