# Wild boundary behaviour of holomorphic functions in domains of $\mathbb{C}^N$

## TL;DR

This paper investigates the boundary behavior of holomorphic functions in complex domains, showing that functions with maximal cluster sets and prescribed boundary limits are prevalent in various function spaces.

## Contribution

It demonstrates the genericity of functions with maximal boundary cluster sets and constructs functions with prescribed boundary limits in complex domains, extending understanding of boundary behaviors.

## Key findings

- Functions with maximal cluster sets are residual, dense, lineable, and spaceable.
- Existence of functions with prescribed boundary limits along smooth curves.
- Results apply to general domains and strictly pseudoconvex domains.

## Abstract

Given a domain of holomorphy $D$ in $\mathbb{C}^N$, $N\geq 2$, 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 space $\mathcal{O}(D)$ of holomorphic functions in $D$. Besides, if $D$ is a strictly pseudoconvex domain in $\mathbb{C}^N$, and if a suitable family of smooth curves $\gamma(x,r)$, $x\in bD$, $r\in [0,1)$, ending at a point of $bD$ is given, then we exhibit a spaceable, densely lineable and residual subset of $\mathcal{O}(D)$, every element $f$ of which satisfies the following property: For any measurable function $h$ on $bD$, there exists a sequence $(r_n)_n \in [0,1)$ tending to $1$, such that \[ f\circ \gamma(x,r_n) \rightarrow h (x),\,n\rightarrow \infty, \] for almost every $x$ in $bD$.