Some open questions and conjectures about visibility and iteration in bounded convex domains in $\mathbb C^N$
Filippo Bracci, Ahmed Yekta Ökten
TL;DR
The note addresses the problem of understanding when visibility and the Denjoy-Wolff property hold for bounded convex domains in $\mathbb{C}^N$ and whether these two dynamical notions are equivalent under minimal regularity. It surveys Kobayashi geometry, complex geodesics, and boundary phenomena to reformulate visibility (including complex visibility) and connects it to the Denjoy-Wolff framework via sequential horospheres and record sequences. It provides partial results and conjectures, notably that visibility and the Denjoy-Wolff property may be equivalent for convex domains and offers detailed analysis for polydiscs and product domains to illuminate coordinate-wise dynamics. The discussions aim to unify iteration theory with boundary geometry and pave avenues for proving the conjectured equivalence in broad convex settings, with implications for holomorphic dynamics in several complex variables.
Abstract
In this note, we propose some open problems and questions about bounded convex domains in $\mathbb C^N$, specifically about visibility and iteration theory.