Maximal cluster sets on spaces of holomorphic functions

Abstract

This is an expository paper where we relate some aspects of the problem of looking for holomorphic functions with maximal cluster sets under the action of operators defined on spaces of holomorphic functions. Some functional generalizations of cluster sets, as well as special spaces of analytic functions, are also considered.

Universal sequences of composition operators

“…Let (r n ) n be a sequence in [0, 1) converging to 1, and let φ n : T → D defined by φ n (z) = r n z. As a particular case of a result stated in [8], there exists a function f ∈ H(D) such that the set {f • φ n : n ∈ N} is dense in the Banach space C(K) of all continuous functions on K (endowed with the sup-norm), for any subset K of T different from T. Such functions enjoy a very singular behaviour near the boundary of T. In general, this type of boundary behaviour is (strictly) wilder than having a dense cluster set along any continuous path to the boundary -another example of erratic boundary behaviour which was considered in several papers, see [19] and the references therein. Note that the sequence (φ n ) n can also be seen as a sequence of holomorphic selfmaps of D and that in this case, it is obviously not universal in the sense considered in the first part of this introduction.…”

Wild boundary behaviour of holomorphic functions in domains of $\mathbb{C}^N$

“…In particular, the previous results apply to O(D) where D is a domain in the complex plane (with L = ∂) and to the space of functions harmonic in domains of R N (with L = ∆), and thus improved earlier works, for instance in the unit disc. We refer to the survey [26] for an overview on the topic of (L)-holomorphic functions with maximal cluster sets before 2008. We shall also mention [7] where the authors are interested in the dense lineability and the spaceability (see below for the definition) in O(D) of the set of so-called universal series (see [24]) with maximal cluster sets along any path to the boundary, and [10] where functions holomorphic in D with a universal property implying the maximality of cluster sets along any such paths are exhibited and studied more specifically.…”