In this paper, 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 T CH . Further, we completely characterize the class of operators T for which M T CH is compact and find it explicitly for several special types of operators. In particular, we present strong evidence that the boundary of M T CH is piecewise analytic in contrast with the boundaries of classical invariant sets occurring in complex dynamics.
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