## Departmental PhD Thesis Exam – Artem Dudko

Everyone is welcome. There will be refreshments served in the Math Lounge before the exam.

DEPARTMENTAL PHD THESIS EXAM

Monday, April 30, 2012, 2:10 p.m., in BA 6183, 40 St. George St.

PhD Candidate: Artem Dudko

Thesis Title: Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets. (http://www.math.toronto.edu/graduate/Dudko-thesis.pdf)

Thesis Abstract:

My thesis consists of two parts. The first part concerns the dynamics of germs with a simple parabolic fixed point at the origin
$F(w)=w+w^2+O(w^3).$ The second part is on Computability of Julia sets. In this talk I will present the results of the first part of the thesis.

Let $F$ be a germ with a simple parabolic fixed point at the origin. It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity
$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$ The dynamics of a germ $f$ can be described using Fatou coordinates. The Fatou coordinates are analytic solutions of
the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution $\tilde{\phi}(z)={\rm const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},$ where $\sum b_jz^{-j}$ is a divergent power series. Using Écalle’s Resurgence Theory I show that $\tilde{\phi}$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J. Écalle and S. Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs
with a parabolic fixed point. I give a new proof of validity of Écalle’s construction.