The Boundary Expansion

The Birkhoff conjecture asks whether the disc is the only convex billiard table with integrable dynamics — where every orbit is confined to a caustic curve. The conjecture remains open, but partial results abound: if a billiard table has certain properties, it must be a disc.

Czudek et al. add another characterization, this time through the formal Lazutkin conjugacy at the boundary.

The Lazutkin conjugacy transforms a billiard map near the boundary (where the ball grazes the edge) into a rotation — a standard form where the dynamics are explicitly integrable. For a general billiard table, this conjugacy exists only as a formal power series; it may not converge. For the disc, it converges and is exact. The characterization identifies the disc by the structure of the coefficients in this formal expansion.

The result is local: it reads the geometry of the billiard table from the asymptotic behavior of orbits that almost miss the boundary. These near-grazing trajectories encode the curvature of the boundary through the Lazutkin series, and the disc’s constant curvature produces a distinctive pattern in the expansion that no other convex table can replicate.

Each new characterization of the disc attacks the Birkhoff conjecture from a different angle — curvature conditions, spectral rigidity, integrable perturbations, and now formal conjugacies. None has closed the conjecture, but together they map the terrain: the disc is recognizable from almost every direction. The conjecture reduces to showing that these many partial views cover all possibilities.