(ii) More generally, one could consider a Riemannian metric with smooth boundary ( M, ∂ M, g): the trajectory starting at x∈∂ M with (inward) unit velocity v will follow the corresponding geodesic until it hits the boundary at x′∈∂ M the reflected (unit) inward velocity v′ is obtained in the following way: the normal component of the hitting velocity instantaneously changes sign, while the tangential one stays unchanged. Besides the study of Birkhoff billiards, very active areas of research focus on the study of polygonal billiards (in particular, rational billiards, whose dynamics can be related to geodesic flows on translation surfaces and Teichmüller theory (e.g. ) or billiards with concave boundary (so-called dispersive billiards) of particular interest as models in statistical mechanics and mathematical physics . (i) The dynamical properties of billiards are strongly related to the geometric properties of its shape. Hereafter, we shall address some of them and describe recent advances towards their solutions. This translates into many intriguing unanswered questions and conjectures that have been the focus of very active research over recent decades. the shape of the billiard table): while it is evident how the shape completely determines the billiard dynamics, a more subtle and intriguing question is to what extent the knowledge of the dynamics allows one to reconstruct the shape of the billiard domain. More remarkably, the dynamics of these systems is profoundly intertwined with their geometric properties (e.g. Mathematically, they offer models in every subclass of dynamical system (integrable, regular, chaotic, etc.) more importantly, techniques initially devised for billiards have often been applied and adapted to other systems, becoming standard tools and having ripple effects beyond the field. Not only are their laws of motion very physical and intuitive, but also billiard-type dynamics are ubiquitous. Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. These conceptually simple models of dynamical systems-yet dynamically very rich and interesting-were first introduced by Birkhoff as paradigmatic examples of Hamiltonian systems, that could be used as a ‘playground’ to shed light, with as little technicality as possible, on some interesting dynamical features and phenomena appearing in the study of their dynamics. In this survey, we provide a concise introduction to convex billiards and describe some recent results, obtained by the authors and collaborators, on the classification of integrable billiards, namely the so-called Birkhoff conjecture.
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