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:

- Iterated Function Systems (IFS's), in which the transformation is the union of a finite number of affine mappings.
- Polynomial systems, where the transformation is a single nonlinear complex polynomial.

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.

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