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:
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].