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

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