We describe dynamical properties of a map defined on the space of rational functions. The fixed points of F are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.
Boros, C., Little, J., Moll, V., Mosteig, E., Stanley, R. A Map on the Space of Rational Functions, Rocky Mountain Journal of Mathematics. vol. 35 (2005) pp. 1861-1880.