An intrinsic notion of visibility and classification of planar Denjoy-Wolff domains

Abstract

In this article, we introduce an intrinsic notion of visibility with respect to the Kobayashi distance that does not require any boundary. We establish localization results for the Kobayashi distance on Kobayashi hyperbolic domains in $\mathbb{C}^d$ that satisfy this intrinsic visibility property. As an application of these localization results, we compare the Busemann horospheres of a domain with those of a subdomain. Recently, Benini and Bracci characterized simply connected planar domains with the Denjoy-Wolff property in terms of Busemann horospheres. More precisely, they proved that a proper simply connected planar domain has the Denjoy-Wolff property if and only if the closure of every Busemann horosphere meets the boundary of the domain in exactly one point. We introduce Condition (M$\star$), a condition of attaching a proper disk to the domain at a boundary point. Using our localization results together with this comparison of Busemann horospheres, we extend the characterization of Benini and Bracci to domains satisfying Condition (M$\star$) at every boundary point, and to domains satisfying it outside a totally disconnected subset of the boundary. In particular, we obtain a characterization for finitely connected domains and for a large class of infinitely connected domains having the Denjoy-Wolff property. These characterizations yield a variety of domains with the Denjoy-Wolff property, which we illustrate through example.