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.