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.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/33985 |
| Date | 11 December 2012 |
| Creators | Dudko, Artem |
| Contributors | Yampolsky, Michael |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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