Bloch functions with wild boundary behaviour in CN

We prove the existence of functions f in the Bloch space of the unit ball BN of CN with the property that, given any measurable function φ on the unit sphere SN, there exists a sequence (rₙ)ₙ, rₙ∈ (0,1), converging to $1$, such that for every w∈ BN, f(rₙ(ζ -w)+w) → φ(ζ) as n→ ∞, for almost every ζ ∈ SN. The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.