Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem

For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set $mathcal U$ in phase space independent of fixed measure, where the set of initial points which lead to collision is $O(mu^frac{1}{20})$ dense as $murightarrow 0$.

Coexistence of periodic-2 and periodic-3 caustics for nearly circular analytic billiard maps

For symmetrically analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of periodic $2,3$ caustics is that the deformation has to be an isometric transformation.