Treschev constructed a billiard given by a formal power series, so that the dynamics near a two-periodic point is formally linearizable. If this example were to converge, then there exists a non-elliptic billiard whose dynamics is integrable on an open set. Thus it would be a counter example (in the context of local integrability) to Birkhoff's conjecture that ellipses are the only integrable billiards.
In this talk, I will explain that the example of Treschev is at least Gevrey. Our method also gives a different proof to Treschev's construction that hopefully will be clarifying. I will speculate on whether this example should converge, providing evidence to both sides. This is based on joint work with Qun Wang.