Linear Fractional Self-Maps of the Unit Ball in $\mathbb{C}^N$

TL;DR

This paper characterizes when a class of generalized linear fractional maps in several complex variables are self-maps of the unit ball, providing new insights into their structure and properties.

Contribution

It introduces a generalized class of linear fractional maps in multiple complex variables and precisely characterizes their conditions to be self-maps of the unit ball.

Findings

Derived new criteria for self-maps of the unit ball in $\

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Abstract

Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in $\mathbb{C}^N$ that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach obtaining numerous new results about this class of maps along the way.