Bloch functions with wild boundary behaviour in $\mathbb{C}^N$

Abstract

We prove the existence of functions $f$ in the Bloch space of the unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable function $\varphi$ on the unit sphere $\mathbb{S}_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \mathbb{B}_N$, $$f(r_n(\zeta -w)+w) \to \varphi(\zeta)\text{ as }n\to \infty\text{, for almost every }\zeta \in \mathbb{S}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.