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. 
Walter Bergweiler and Alexandre Eremenko