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Geometric properties of linear fractional maps

Linear fractional maps in several variables generalize classical linear fractional maps in the complex plane. In this paper, we describe some geometric properties of this class of maps, especially for those linear fractional maps that carry the open unit ball into itself. Some properties of the derivative of linear fractional maps at their fixed points are described. For those linear fractional maps that take the unit ball into itself, we determine the minimal set containing the open unit ball on which the map is an automorphism. Finally, when (p is a linear fractional map, we describe the linear fractional solutions, f, of Schroeder's functional equation f o φ = Lf.