Meromorphic functions with linearly distributed values and Julia sets of rational functions

If the preimage of a four-point set under a meromorphic function belongs to the real line then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle. If the full preimage of a two-point set under a rational function belongs to the real line then the function maps the real line to a circle (on the Riemann sphere). We prove an analog of this simple fact for transcendental functions meromorphic in the plane. Theorem 1. Let f be a meromorphic function such that the preimage of three points belongs to the real line. Then f maps the real line into a circle, unless f(z) = L ( 1− ei(c1z−b1) 1− ei(c2z−b2) ) , (1) where L is a fractional-linear transformation, and cj , bj are real numbers. ∗Supported by the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003, the EU Research Training Network CODY, and the ESF Research Networking Programme HCAA. †Supported by NSF grant DMS-0555279.