On convex planar billiards, Birkhoff Conjecture and whispering galleries
Vortrag von Prof. Dr. Vadim Kaloshin Datum: 09.02.17 Zeit: 10.10 - 10.50 Raum: Y27H28
Abstract: A mathematical billiard is a system describing the inertial motion of a point
mass inside a domain with elastic reflections at the boundary. In the case of convex planar
domains, this model was first introduced and studied by G.D. Birkhoff, as a paradigmatic
example of a low dimensional conservative dynamical system.
A very interesting aspect is represented by the presence of `caustics', namely curves inside
the domain $\Omega$ with the property that a trajectory, once tangent to it, stays tangent
after every reflection (as on the left Figure). Besides their mathematical interest, these objects
can explain a fascinating acoustic phenomenon, known as "whispering galleries", which can
be sometimes noticed beneath a dome or a vault.
The classical Birkhoff conjecture states that the only integrable billiard, i.e., the one having
a region filled with caustics, is the billiard inside an ellipse. We show that this conjecture holds
true for perturbations of ellipses preserving many caustics. This consists of two main steps:
perturbations of a circle (joint with A. Avila—J. De Simoi) and perturbations of an ellipse
(joint with A. Sorrentino). In a somewhat different direction we prove this conjecture for
perturbations of the circle preserving not so many caustics (joint w G. Huang—A. Sorrentino).