Alexey Glutsyuk
(École normale supérieure de Lyon)
On periodic orbits in complex planar billiards
A conjecture of Victor Ivrii (1980) says that in every
billiard with smooth boundary the set of periodic
orbits has measure zero. This conjecture is closely
related to a conjecture of Hermann Weyl (1911) from the
spectral theory. The particular case of Ivrii's
conjecture for triangular orbits was proved in dimension
two by M. Rychlik (1989), several other mathematicians,
and in arbitrary dimension by Ya. Vorobets (1994). The
case of quadrilateral orbits in planar billiards has been
recently treated in our joint paper with
Yu. Kudryashov. A new approach to Ivrii's conjecture is to
study complex billiards. We will discuss the
complexified version of Ivrii's conjecture for
reflections with respect to complex planar analytic
curves. It appears that thus complexified Ivrii's
conjecture is false, and it would be interesting to
classify the counterexamples. We will show that the
only "nontrivial" counterexamples with four reflections
are formed by couples of confocal conics. This result
has an application to an analogue of the real Ivrii's
conjecture: the invisibility problem. If the time
allows, we will discuss a small result concerning odd
number of reflections.