Bounded holomorphic functions and maps with some boundary behavior (Q1410274)

Let \(\Omega\) be a domain of \(\mathbb C^n\). As a study of boundary behavior of functions and maps in \(\Omega\), the author considers the cluster set along segments terminating at a boundary point of \(\Omega\). It is proved that under some conditions, there exists a bounded holomorphic function defined in \(\Omega\) whose arbitrary linear cluster set at every point of a discrete subset of the boundary \(\partial\Omega\) contains a closed disk of positive radius, or a closed annulus of positive thickness. This result gives some type of counter part of Fatou's theorem of arbitrary dimension. The existence of a bounded holomorphic map with wild boundary behavior is shown.