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SUMMARY:Illya Koval (Institute of Science and Technology Austria) -- Billi
ard tables with analytic Birkhoff normal form are generically Gevrey diver
gent
DTSTART:20240319T100000Z
DTEND:20240319T110000Z
DTSTAMP:20240529T113200Z
UID:indico-event-11405@indico.math.cnrs.fr
DESCRIPTION:The problem of the existence of an analytic normal form near a
n equilibrium point of an area-preserving map and analyticity of the assoc
iated coordinate change is a classical problem in dynamical systems going
back to Poincaré and Siegel. One important class of examples of area-pre
serving maps consists of the collision maps for planar billiards. Recently
\, Treschev discovered a formal bi-axially symmetric billiard with locally
linearizable dynamics and conjectured its convergence. Since then\, a Gev
rey regularity for such a billiard was proven by Q. Wang and K. Zhang\, bu
t the original problem about analyticity still remains open. We extend
the class of billiards by relaxing the symmetry condition and allowing con
jugacies to non-linear analytic integrable normal forms. To keep the forma
l solution unique\, odd table derivatives and the normal form are treated
as parameters of the problem. We show that for the new problem\, the serie
s of the billiard table diverge for general parameters by proving the opti
mality of Gevrey bounds. The general parameter set is prevalent (in a cert
ain sense has full measure) and it contains an open set. In order to prove
that on an open set Taylor series of the table diverges we define a Taylo
r recurrence operator and prove that it has a cone property. All solutions
in that cone are only Gevrey regular and not analytic.\n\nhttps://indico.
math.cnrs.fr/event/11405/
LOCATION:salle I (laboratoire Dieudonné)
URL:https://indico.math.cnrs.fr/event/11405/
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