# 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].

dieter@cs.wisc.edu