My research is in the area of holomorphic
dynamics, particularly on the structure of
polynomial Julia sets. Here is a famous Julia
set, the Douady Rabbit.
I am particularly interest in polynomials of
degree larger than 2. Below is the Julia set
of a cubic polynomial that I like. I call it A Family of Rabbits.
I introduced a method to code the dynamics of
a polynomial with disconnected Julia set using the
combinatorial system of a tree with dynamics. To the
left is the Julia set of z²-3, in the center some
equipotentials are highlighted, and to the right its
tree with dynamics.
I have found a new family of meta-Fibonacci numbers. They occur as the return times of polynomials.
I have study the interaction of Brownian motion in the complex plane and Julia sets by coding the Brownian motion on the tree with dynamics.
Articles:
Dynamics of Polynomials with Disconnected Julia Sets."Discrete and Continuous Dynamical Systems - Series A, Volume 9, Number 4, July 2003.
http://aimsciences.org/journals/displayPapers.jsp?comments=&pubID=30&journID=1&pubString=Volume:%209,%20Number:%204,%20July%202003
(Off-prints available upon request).
A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions.
Journal of Integer Sequences, Volume 9, Issue 1, Article 06.1.8, 2006.
http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Emerson/emerson6.pdf
Return Times of Polynomials as meta-Fibonacci Numbers.
Conformal Geometry and Dynamics, Volume 12, October 2008.
http://www.ams.org/ecgd/2008-12-10/S1088-4173-08-00183-5/
Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets.
Under revision.
http://arxiv.org/abs/math.DS/0609044
On Yoccoz Return Functions.
Available on ArXiv
http://arxiv.org/abs/0908.3326