Vol. 298, No. 2, 2019

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

Joshua Strong

Vol. 298 (2019), No. 2, 471–481
Abstract

The classification of bounded domains in 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 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 Ω n is a bounded convex domain with C boundary. Suppose there exists φ Aut(Ω) and p Ω such that for the sequence of iterates {φj} Aut(Ω) we have φj(p) x Ω 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
Authors
Joshua Strong
Whittier College
Whittier, CA
United States