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We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points $A,B$ such that no finite set of points can block all billiard trajectories from $A$ to $B$.
On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards
Following a recent paper by Baryshnikov and Zharnitsky, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has empty interior.
The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of geometrically meaningful discrete integrable maps.
The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call twisted polygons. In this note we describe our recent work on the pentagram map, in which we find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.
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