A New Characterization of Line-To-Line Maps in the Upper Plane

Baokui Li, Yuefei Wang

The characterization of typical maps in a domain of a given space is a much harder problem than that in the whole space. In this paper, by using methods of hyperbolic and affine geometry, we give a new characterization of line-to-line maps in the upper plane. We show that a line-to-line surjection is either an affine transformation, or a composition of an affine transformation and a $g$-reflection. Moreover, we prove that the composition of two $g$-reflections with the same boundary is an affine transformation.