CRM: Centro De Giorgi
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Local holomorphic dynamics

seminar: Resurgence of parabolic curves in C2

speaker: David Sauzin (CNRS)

abstract: Joint work with Vassili Gelfreich (Warwick University) We study parabolic germs of holomorphic transformations of $(C2,0)$ with unipotent linear part, under some non-degeneracy hypothesis on the jet of order 2. They always admit invariant curves which may be called stable and unstable manifolds, and can be obtained as Borel sums of a single formal expansion involving powers of~$z{-1}$ and~$z{-1}\log z$, where~$z$ is a large variable. This formal curve is in fact the separatrix of the formal infinitesimal generator of the parabolic germ, i.e. a formal vector field with nilpotent linear part, the time-$1$ map of which coincides with the given germ.

Generically, the formal separatrix is divergent and its two sums are not the analytic continuation one of the other; the leading order of their difference, which is exponentially small, is determined by a pair of complex constants. We analyse this phenomenon in the framework of Resurgence theory and prove that these constants depend analytically on parameters. They vanish for the time-1 map of a holomorphic vector field but not for a generic germ. We also study the "formal integral", which to corresponds to the invariant foliation induced by the formal infinitesimal generator, and derive the so-called Bridge Equation in order to describe completely the resurgent structure.


timetable:
Thu 25 Jan, 12:20 - 13:20, Aula Dini
documents:

sauzin.pdf



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