Linearization of transseries and transserial maps in a neighborhood of a singular point

Jean-Philippe Rolin
Institut de Mathématiques de Bourgogne, Dijon

Date(s) : 06/05/2021   iCal
14 h 00 min - 15 h 00 min

It is known since Schröder (1873) that in order to study the iterations of a function in the neighborhood of a fixed point (which can be seen as a singularity of the system in this context), the best way, when possible, is to linearize the function, that is to find a coordinate in which the function becomes a linear map. The linearization problem has been solved by Koenigs for holomorphic maps in one variable under the hypothesis of « hyperbolicity ». This result has been extended to several category of differentiable functions.

Our goal is to show a version of this result in the framework of « transseries », which are formal power series involving the exponential and logarithm functions. The proof investigates valuations and various topologies on the class of transseries. This result has an application in the dynamics of so-called Dulac maps, which play a important role in a celebrated conjecture on planar polynomial vector fields. Joint with D. Peran, M. Resman and T. Servi.


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