Open Problems about Birkhoff Billiards
Every Frenet curve in the plane defines a Birkhoff billiard: a strictly convex curve. This can be described nicely in complex notation. The Frenet equations are now directly integrated using a complex function r'(t) = rho(t) exp(i t) which plays the role of the curvature matrix in so(2) when writing it out in the real.. Integrating gives the curve producing explicit formulas for trig polynomials rho(t). We can also describe the evolute nicely as attaching rho(t) i exp(i t) to r(t). There are HIlbert spaces of periodic functions which correspond to curves of constant width. This is therefore a rich playground to explore open problems about Birkhoff billiards. One does not know whether there are integrable cases different from ellipses, one does not have examples of positive entropy, one does not know about whether the table is defined by the length spectrum or equivalently by the Dirichlet spectrum. One has also no example beside the ellipse of a billiard map that is algebraic in suitable coordinates. I had used the frame work in 1998 to get fractal evolutes or fractal striction curves of ruled surfaces or fractal caustics for metrics on the 2-sphere. Fractal means here just nowhere differentiable except for finitely many points - hedgehogs. We do not have yet the Hausdorff dimension.
See this paper https://people.math.harvard.edu/~knil...