On weak Wolff--Denjoy theorem for certain non-convex domains

TL;DR

This paper identifies a class of complex domains where holomorphic self-maps either have fixed points or diverge, and describes fixed point sets for specific domains like the symmetrized bidisc.

Contribution

It introduces a new class of non-convex domains in complex space with fixed point or divergence properties for holomorphic self-maps, including explicit descriptions for key examples.

Findings

Symmetrized bidisc, tridisc, tetrablock, pentablock belong to the class.

Holomorphic self-maps either have fixed points or diverge.

Fixed point sets are explicitly described for certain domains.

Abstract

In this paper, we provide a class of domains in $\mathbb{C}^3$, such that every holomorphic self-map of that domain either has a fixed point or the sequence of iterates is compactly divergent. In particular, it follows that the symmetrized bidisc, symmetrized tridisc, tetrablock, pentablock are in the aforementioned class of domains. We also give a description of the fixed point set of a holomorphic self-map of the symmetrized bidisc and tetrablock. For the symmetrized bidisc, given a holomorphic self-map such that the sequence of iterates is compactly divergent, we also provide a description of its target set.