Research on Complex Dynamical Systems
Dieter van Melkebeek

A complex dynamical system is a continuous transformation of the complex plane. An intriguing question is what happens with points or larger subsets of the complex plane when we keep applying the transformation. Well-studied examples include:

Polynomial Systems

The dynamical behavior of a nonlinear complex polynomial P can be quite intricate. The Julia set of P plays a crucial role in the study of the dynamics. It is the boundary of the set of points z on which the iterates of P converge to infinity, i.e., on which the modulus of the points z, P(z), P(P(z)), ... goes to infinity. The famous Mandelbrot set consists of those complex numbers c for which the Julia set of the polynomial Pc that maps z to z2+c, is connected. The complement of the Mandelbrot set contains precisely those complex numbers c for which the iterates of Pc evaluated in the origin go to infinity. The rate of convergence is typically used as the index in a color map to obtain nice pictures of the Mandelbrot set.

The iterates of P (up to an additive constant) turn out to be orthogonal with respect to a natural measure on the Julia set of P, namely the logarithmic equilibrium measure, which describes the equilibrium distribution of electric charges on a cylindrical conductor with the Julia set as its cross section. This orthogonality property has applications in various areas and has been well-studied. The orthogonality of the classical Chebyshev polynomials forms a particular instantiation.

Under supervision of A. Bultheel, I have worked on several aspects of general orthogonality relations between the iterates of a complex polynomial. In particular, I looked at the existence and construction of orthogonal polynomials of the missing intermediate degrees between those of the iterates, and at recurrence relations between them [paper]. I also investigated the structure of P-invariant measures, not just for complex polynomials P, but for rational functions P on the Riemann sphere [paper]. The above mentioned logarithmic equilibrium measure on the Julia set of P can be characterized as a very special P-invariant measure on the complex plane, namely the balanced one.

Together with M. Van Barel, I wrote a survey paper (in Dutch) on Julia sets and the Mandelbrot set.

Iterated Function Systems and Image Compression

Compared to polynomial systems, the dynamics of IFS's are fairly simple. The interesting IFS's, the so-called hyperbolic ones, all have an attractor, i.e., a subset of the plane to which any nonempty compact subset converges under the iterates. The attractors have a fractal structure which is sufficiently rich to approximate a lot of natural objects. This fact can be used as the basis for lossy image compression.

In my undergraduate thesis, I investigated the feasibility of using polynomial systems for image compression. I also compared some variants of IFS's [paper].