On analytic projective billiards with open sets of triangular orbits

This talk will present a generalization of billards called projective billards. In such billards, the law of reflexion is not defined by the usual orthogonal symmetries with respect to the tangent lines of the billard tables. Instead, the curves defining the billard tables are endowed with a field of lines, giving place to another reflexion law at each point of the borders. Playing on these tables, we can therefore investigate the same questions as for the usual billards, and for example try to answer Ivrii's conjecture: are there billard tables on which one can find a two-dimensionnal set of periodic orbits? Even more, is it possible to classify such tables? I will present a result for triangular periodic orbits, and try to show how analytic geometry can be useful in such theory.