# MAT 655: Introduction to transcendental dynamics

# Spring 2016

# Christopher Bishop

Office: 4-112 Mathematics Building

Phone: (516)-632-8274

Dept. Phone: (516)-632-8290

FAX: (516)-632-7631

TuTh 1:00pm to 2:20pm, Physics P-124

I am trying to write some lecture notes to go along with
the class. You can find the current version
here ,
but they are incomplete and very rough.

## Introduction to the course

I was planning to start
with some basic results about the iteration of entire functions:
Eremenko's theorem that the escaping set is non-empty,
Baker's theorem that multiply connected Fatou components
can exist and are wandering domains, Misiurewic's theorem
that the Julia set of e^z is the whole plane and the fact that
repelling fixed points are always dense in the Julia set.
Along the way we will review basic facts of geometric
function theory that are needed: the hyperbolic metric, the
uniformization theorem, Koebe's theorem, Piacard's theorems,
and the Ahlfors Isands theorem.

Depending on time and the interest of participants, I will
then turn to computing the dimension of certain Julia sets.
Baker's theorem mentioned above implies the Julia set of
an entire function always contains a non-trival continuum,
hence it has Hausdorff dimension at least one. It is
fairly easy to build examples with Hausdorff dimension 2,
somewhat harder to get values between 1 and 2, and only a recent
result that dimension 1 can be attained. I also hope to
discuss the construction of functions in the Eremenko-Lyubich
class (bounded singular set), and construct some `fun' examples,
such as the counterexample to the strong Eremenko conjecture.

Webpage for
a previous course on this topic, MAT 627, Spring 2013. This page
gives a brief introduction to the topics covered in
both classes and links to a number of relevant papers.

Send the lecturer (C. Bishop) email at:
bishop - at - math.sunysb.edu

Email all MAT 655 participants
(Bishop, Adams, Lazebnik, Younsi, Arfeux, Bedford,
Mukherjee, Carsten, Strangberg, Goldberg, Winckler, Albrecht )