Office: 4-112 Mathematics Building
Dept. Phone: (516)-632-8290
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.
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.
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