Finiteness of the fixed point sets of automorphisms

TL;DR

This paper studies the fixed point sets of automorphisms in complex bounded domains, showing that under certain conditions, fixed points are finite and providing bounds on their number.

Contribution

It extends classical results from one complex variable to higher dimensions, establishing finiteness and uniform bounds for fixed points of automorphisms.

Findings

Nontrivial automorphisms in one variable have at most two fixed points.

Discreteness of fixed point sets in higher dimensions implies finiteness.

A uniform bound exists for fixed points of automorphisms in certain compact subgroups.

Abstract

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be discrete. We show, under natural extension hypotheses, that discreteness forces finiteness. We also obtain a uniform bound for the number of fixed points of automorphisms in compact subgroups whose elements admit such extensions.