Differences and similarities between multiplicatively and additively universal entire functions and universality properties of compositionally non-normal families of holomorphic functions
Final Report Abstract
During my research stay in Sevilla, three mathematical works have been completely developed. The first work introduces the notion of simultaneous universality concerning operators having orbits that simultaneously approximate any given vector. This approach is related to the well-known concepts of universality and disjoint universality. As simultaneous universality is a weaker approximation property than disjoint universality, our paper furnishes several examples of finite families of operators being simultaneously but not disjointly universal. Mainly, the settings of sequence spaces and of spaces of holomorphic functions are considered. The most important result of the work is given by the simultaneous hypercyclicity criterion which, analogously to the already known hypercyclicity and disjoint hypercyclicity criteria, provides e sufficient conditions under which continuous linear operators on a separable Fréchet space are simultaneously hypercyclic. In the second work, universality properties of generalized backward shifts on sequence spaces are studied. The considered shift operators are associated to certain continuous selfmappings of a topological space, and simultaneous and disjoint universality of finite families of generalized backward shifts are investigated with respect to dynamical properties of the generating selfmappings. If the considered selfmappings are given by rational functions of degree at least two, the mixing property as well as simultaneous and disjoint universality properties of the corresponding backward shift operators are characterized in terms of the Julia set of the given rational functions. The third work establishes connections between the property that a countable family of meromorphic functions is a universal family, i.e. that restrictions of this family to suitable open or compact subsets are dense families in the corresponding function spaces, and the property that it is a non-normal family. In particular, relations between compositional universality and non-normality of such families are provided, and connections concerning non-normality of families at all points of suitable subsets and possible limit functions of sequences in the family are furnished. It is proved that, in general, compositional universality is a strictly stronger property than nonnormality, and it is pointed out that even in case of a strong kind of non-normality of a family of holomorphic functions it might be the case that it does not possess any universality properties at all. However, the property of being not normal at each point of a given subset implies that, under suitable natural additional conditions, the family possesses a certain kind of universality property on the considered subset.
Publications
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On connections between universality and non-normality
L. Bernal-González, A. Jung, J. Müller
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Mixing, simultaneous universal and disjoint universal backward Φ-shifts, J. Math. Anal. Appl. 452 (2017), 246-257
Jung, A.
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Simultaneous universality
L. Bernal-González, A. Jung
