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Dynamics of Holomorphic Maps: Resurgence of Fatou coordinates, and Poly-time Computability of Julia Sets

The present thesis is dedicated to two topics in Dynamics of
Holomorphic maps. The first topic is dynamics of simple parabolic
germs at the origin. The second topic is Polynomial-time
Computability of Julia sets.\\

Dynamics of simple parabolic germs. Let $F$ be a germ with a
simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ 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.
Fatou coordinates are analytic solutions of the equation
$\phi(f(z))=\phi(z)+1.$ This equation has a formal solution
\[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where
$\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show
that $\tilde$ 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.~\'Ecalle and S.~Voronin independently constructed a
complete set of invariants of analytic conjugacy classes of germs
with a parabolic fixed point. We give a new proof of validity of
\'Ecalle's construction.
\\
Computability of Julia sets. Informally, a compact subset of
the complex plane is called \emph if it can be
visualized on a computer screen with an arbitrarily high precision.
One of the natural open questions of computational complexity of
Julia sets is how large is the class of rational functions (in a
sense of Lebesgue measure on the parameter space) whose Julia set
can be computed in a polynomial time. The main result of Chapter II
is the following: Theorem. Let $f$ be a rational
function of degree $d\ge 2$. Assume that for each critical
point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain
either a critical point or a parabolic periodic point of $f$. Then
the Julia set $J_f$ is computable in a polynomial time.

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OTU.1807/33985
Date11 December 2012
CreatorsDudko, Artem
ContributorsYampolsky, Michael
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
Languageen_ca
Detected LanguageEnglish
TypeThesis

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