Limits of quadratic rational maps with degenerate parabolic fixed points of multiplier $\exp{2piiq}\rightarrow 1}$.
speaker: Adam Epstein (University of Warwick)abstract: (Joint work with Xavier Buff and Jean Ecalle) We study the bifurcation loci in families $$z\mapsto \exp{-2piiq}(z+A+\frac{1}{z})$$ of quadratic rational maps with parabolic fixed point at infinity, and consider the limit as $q\rightarrow 1$. This limit is considerably richer than the the bifurcation locus for the limiting family $$z\mapsto z+ A + \frac{1}{z}.$$ This enrichment is explained in terms of "parabolic implosion", and its fine structure is a reflection of properties of the Ecalle-Voronin invariants for the limiting family. We will focus on one of the simplest features, namely the distribution of parameters for which these invariants have vanishing first coefficient. Supporting estimates may be obtained in two different way; our approach will exploit geomerry and dynamics, but we will touch on Borel-Laplace summation as well.
timetable:
Mon 12 Oct, 14:45 - 15:45, Aula Dini
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