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SUMMARY:On periodic orbits in complex billiards
DTSTART;VALUE=DATE-TIME:20141001T120000Z
DTEND;VALUE=DATE-TIME:20141001T130000Z
DTSTAMP;VALUE=DATE-TIME:20190819T012704Z
UID:indico-contribution-947@indico.math.cnrs.fr
DESCRIPTION:Speakers: Alexey GLUTSYUK (Lyon and Moscow)\nA conjecture of V
ictor Ivrii (1980) says that in every billiard with smooth boundary the se
t of periodic orbits has measure zero. This conjecture is closely related
to spectral theory. Its particular case for triangular orbits was prov
ed by M. Rychlik (1989)\, Ya. Vorobets (1994) and other mathematician
s\, and for quadrilateral orbits in our joint work with Yu. Kudryashov (20
12). \nWe present a new approach to planar Ivrii's conjecture for billia
rds with piecewise-analytic boundary: to study its complexified version wi
th reflections from holomorphic curves. The direct complexification of Iv
rii's conjecture is false in general. \nIt would be interesting for real
applications to classify the counterexamples: complex billiards with open
sets of periodic orbits of a given period. We will show that the only "
nontrivial" counterexamples with four reflections are formed by couples of
confocal conics. We will discuss a small result concerning odd number of
reflections. We provide applications of these results to real billiards\,
including Plakhov's Invisibility Conjecture and Tabachnikov's commuting bi
lliard problem.\n\nhttps://indico.math.cnrs.fr/event/201/contributions/947
/
LOCATION:International Center for Theoretical Physics Adriatico Building\,
Kastler Lecture Hall
URL:https://indico.math.cnrs.fr/event/201/contributions/947/
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