Iterated automorphism orbits of bounded convex domains in $\mathbb C^{n}$

Joshua Strong

Vol. 298 (2019), No. 2, 471–481

DOI: 10.2140/pjm.2019.298.471

Abstract

The classification of bounded domains in ${ℂ}^n$, with $n>1$, is related to the geometric properties of the boundary. A conjecture of Greene and Krantz relates the geometry of the boundary with the group of biholomorphic self mappings of the domain. The Greene–Krantz conjecture, if true, can tell us much about the classification of smoothly bounded domains in ${ℂ}^n$. Much work has been done to attempt to solve this conjecture, though it has yet to be proved or disproved. However, there are numerous partial results which support the conjecture. In this paper, we prove a special case of the conjecture:

Theorem: Suppose $\Omega \subset {ℂ}^n$ is a bounded convex domain with $C^{\infty }$ boundary. Suppose there exists $\phi \in Aut\left(\Omega \right)$ and $p\in \Omega$ such that for the sequence of iterates $\left\{{\phi }^j\right\}\subset Aut\left(\Omega \right)$ we have ${\phi }^j\left(p\right)\to x\in \partial \Omega$ nontangentially. Then $x$ is of finite type.

Keywords

several complex variables, Greene Krantz conjecture, finite type, convex domains, automorphism orbit accumulation points, iterated automorphisms

Mathematical Subject Classification 2010

Primary: 32-02

Milestones

Received: 5 July 2017

Revised: 16 July 2018

Accepted: 10 August 2018

Published: 8 March 2019