We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves in the complex plane which are invariant under a rational function.
"Decompositions of rational functions over real and complex numbers and a question about invariant curves." Illinois J. Math. 59 (4) 825 - 838, Winter 2015. https://doi.org/10.1215/ijm/1488186011