Differential equations and complex dynamics

The main purpose of the project is to combine, cross-fertilize and substantially develop two existing areas of mathematics, linear ordinary differential equations with complex coefficients with the special emphasis on the Pólya-Schur theory and the theory of iterations of rational functions with the emphasis on their Julia sets.The main question under consideration is called the inverse problem in the Pólya-Schur theory asking the following.

Given a linear univariate differential operator L with polynomial coefficients, we say that a closed subset S of the complex plane is L-invariant if for any polynomial p(x) with roots in S, L(p(x)) either vanishes identically or has all its roots in S.Problem:   Given an arbitrary differential operator L as above, describe a sufficiently large class of L-invariant closed subsets.

Ultimately, characterize all non-trivial L-invariant closed subsets of the complex plane.It turns out that for a large class of operators L there exists a unique closed and minimal under inclusion L-invariant set M^L. The ultimate goal is to describe M^L for as many different types of operators as possible.