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Nils Hemmingsson: Linear differential operators and their Hutchinson-invariant sets

Time: Fri 2023-03-17 15.00 - 17.00

Location: Cramer room, Albano building 1 and Zoom

Video link: Meeting ID: 655 1635 8276

Doctoral student: Nils Hemmingsson

Opponent: Andrei Agrachev (SISSA)

Supervisor: Boris Shapiro

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In this licentiate thesis we consider problems related to what we call Hutchinson-invariance, which is a form of invariance for sets in the com- plex 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 felds 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 prop- erties 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.

Read the thesis on DiVA