In this licentiate thesis we consider problems related to what we call Hutchinson invariance, which is a form of invariance for sets in the complex plane or the Riemann sphere with respect to the action of special differential operators.

In the introductory chapter, we provide a background on Hutchinson invariance, explain how it relates to other problems in dynamical systems and why it is an interesting subject of study. In particular, we relate it to the Pólya-Schur theory, rational vector fields as well as iterations of rational functions and algebraic correspondences.

Paper I is joint with my principal and secondary supervisors, Boris Shapiro and Per Alexandersson. It studies what we call continuous Hutchinson invariance for first order differential operators, which is a special form of invariance for sets in the complex plane. We investigate a variety of properties of these invariant sets. For instance, we describe when there exists a minimal under inclusion continuously Hutchinson invariant set, when it is compact, when it coincides with the whole complex plane and when it equals a line or line-segment.

Paper II is entirely written by myself. It studies what we call merely Hutchinson invariance, which is a concept that is closely related to that studied in Paper I. In this second paper, we among other things find that there exists a minimal under inclusion Hutchinson invariant set for a large class of linear operators, and that in this case, this set is bounded and perfect.