Wild boundary behaviour of homomorphic functions in domains of C^N

Abstract

Given a domain of holomorphy D in C , 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 spaceO(D) of holomorphic functions in D. Besides, if D is a strictly pseudoconvex domain in C , 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 (rn)n ∈ [0, 1) tending to 1, such that f ◦ γ(x, rn)→ h(x), n→∞, for almost every x in bD.