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.

Douady's 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.


More trees 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.


Dynamics of Polynomials with Disconnected Julia Sets."
Discrete and Continuous Dynamical Systems - Series A, Volume 9, Number 4,  July 2003.,%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.

Return Times of Polynomials as meta-Fibonacci Numbers.
Conformal Geometry and Dynamics, Volume 12, October 2008.

Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets.
Under revision.

On Yoccoz Return Functions.
Available on ArXiv