A weak notion of visibility, a family of examples, and Wolff--Denjoy theorems

TL;DR

This paper explores a weak form of visibility in complex domains related to the Kobayashi distance, establishing new Wolff--Denjoy theorems and constructing examples of visibility domains that are not Goldilocks domains.

Contribution

It introduces a new, weaker notion of visibility for complex domains, provides conditions for such domains, and constructs examples that distinguish these from Goldilocks domains.

Findings

Some Wolff--Denjoy theorems hold under weak visibility conditions.

Constructed domains are visibility domains but not Goldilocks domains.

Visibility relates to the Kobayashi distance and affects complex dynamics.

Abstract

We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $\mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $\Omega$ in $\mathbb{C}^n$, $n\geq 2$, it is, in general, not even known whether the metric space $(\Omega,{\sf k}_{\Omega})$ (where ${\sf k}_{\Omega}$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems.