Analytic rigidity and symbolic dynamics for two-centre billiards

We establish a sharp rigidity–chaos dichotomy for planar two-centre billiards, motivated by a natural analogue of the Birkhoff–Poritsky conjecture: the only tables integrable at every energy should be ellipses confocal with the two centres. Let $\mathrm{\Omega }$ be a bounded domain with ${\mathcal{C}}^1$ boundary containing the segment joining the centres. At every fixed energy $h\ge 0$, if $\mathrm{\partial }\mathrm{\Omega }$ is not a confocal ellipse, we construct billiard trajectories that shadow the stable and unstable manifolds of the collision–reflection orbit and realise arbitrarily prescribed sequences of sufficiently large winding numbers around the segment. This yields an invariant set semiconjugate to the full shift on a countable alphabet, periodic trajectories with prescribed finite itineraries, and compact invariant subsystems with arbitrarily large topological entropy. If, in addition, $\mathrm{\partial }\mathrm{\Omega }$ is real-analytic, every real-analytic function on the fixed-energy phase space $M_h$ that is invariant under the billiard map is constant. This establishes the real-analytic form of the two-centre Birkhoff–Poritsky conjecture throughout the non-negative-energy regime.