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 $\ddot{I}†$ 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 $w\in {\mathbb{B}}_N$, $$f(r_n(\hat{I}\P -w)+w)\to \ddot{I}†(\hat{I}\P )\text{ as }n\to \infty \text{, for almost every }\hat{I}\P \in {\mathbb{S}}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.