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

Abstract

Given a domain of holomorphy D in C N , N ≥ 2, we show that the set of holomorphic functions in D whose cluster sets along any finite length paths to the boundary of D is maximal, is residual, densely lineable and spaceable in the space O(D) of holomorphic functions in D. Besides, if D is a strictly pseudoconvex domain in C N , and if a suitable family of smooth curves γ(x, r), x ∈ bD, r ∈ [0, 1), ending at a point of bD is given, then we exhibit a spaceable, densely lineable and residual subset of O(D), every element f of which satisfies the following property: For any measurable function h on bD, there exists a sequence (r n ) n ∈ [0, 1) tending to 1, such thatfor almost every x in bD.