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.

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


 

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