| Summary: | This thesis deals with two different topics. In the first part we study Lagrange spectra of Veech translation surfaces, which are a generalisation of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. We start from the concrete example given by the surface obtained glueing a regular octagon. We use the coding developed by Smillie and Ulcigrai for the surfaces obtained glueing the regular 2ɳ-gons to code geodesics in the Teichmiiller disk of the octagon and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets. In particular this yields an estimate on the beginning point of the Hall ray. Generalising the approach of the octagon, in a joint work with Luca Marchese and Corinna Ulcigrai, we prove the existence of a Hall ray for the Lagrange spectrum of any Veech translation surface. In this case, we use the boundary expansion developed by Bowen and Series. In the second part, we construct exceptional examples of ergodic vertical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are Z²-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed. iii
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