Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points

Abstract

There are known conformal self-mappings of the fundamental domains of analytic functions via Möbius transformations. When two adjacent fundamental domains have a straight line or an arc of a circle as a common boundary, the Schwarz symmetry principle can be applied for one of those mappings and what we obtain is a conformal self-mapping of the union of those domains in which each one of the domains is mapped onto itself. Repeating this operation until the whole plane is exhausted, we obtain a conformal self-mapping of the complex plane in which every fundamental domain is conformally mapped onto itself. We prove in this paper that this is true for any analytic function. Since the self-mappings of fundamental domains have each one at least one fixed point, ultimately, for the self-mapping of the complex plane, we obtain at least as many fixed points as is the number of fundamental domains. When dealing with a rational function, this number is finite, otherwise we obtain infinitely many fixed points. Computer experimentation allows the illustration of these concepts for most of the familiar classes of analytic functions. There are known applications of the Möbius transformations in physics via the Lorentz group. Relating those application to the present work may contribute to the advancement of the knowledge in that field.