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Nils Hemmingsson: First order differential operators and Hutchinson-invariant sets

Time: Thu 2022-04-07 13.00

Location: KTH, Room 3721, Lindstedtsvägen 25

Participating: Nils Hemmingsson (Stockholm University)

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Abstract: We initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M. Further, we completely characterize the class of operators T for which M is compact and find it explicitly for several special types of operators. In particular, we present strong evidence that the boundary of M is piecewise analytic in contrast to the boundaries of classical invariant sets occurring in complex dynamics. This is joint work with Per Alexandersson and Boris Shapiro.