On the formally linearizable billiard of Treschev

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.