On the boundary orbit accumulation set for a domain with non-compact automorphism group

TL;DR

This paper investigates the boundary accumulation set for automorphism groups of certain complex domains, showing it is compact, connected if large enough, and related to the Shilov boundary.

Contribution

It establishes the compactness and connectedness properties of the boundary accumulation set for domains with non-compact automorphism groups, linking it to the Shilov boundary.

Findings

The boundary accumulation set is always compact.

If it contains at least three points, it is connected.

The set is a subset of the Shilov boundary.

Abstract

For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is a compact subset of $\partial D$ and, if $S(D)$ contains at least three points, it is connected and thus has the power of the continuum. We also discuss how $S(D)$ relates to other invariant subsets of $\partial D$ and show in particular that $S(D)$ is always a subset of the \v{S}ilov boundary.