Irene de Blasi (University of Torino)
Abstract: We investigate the integrability of Kepler billiards, that is, mechanical
billiard systems in which a particle moves under the influence of a gravitational center
and reflects elastically at the boundary of a strictly convex planar domain. Our main
result establishes that, except possibly for one location of the gravitational center,
analytic integrability at high energies occurs only when the domain is an ellipse and
the center is placed at one of its foci. This provides a partial affirmative answer to a
Keplerian analogue of the classical Birkhoff-Poritsky Conjecture.
Our approach is based on the construction of symbolic dynamics at high energies, whose existence is ensured provided suitable conditions on the boundary are satisfied, and implies positive topological entropy.
Joint work with S. Baranzini, V, Barutello and S. Terracini