Rational Functions with a Polynomial Iterate Original Research Article

Let φ(z)set membership, variantimage(z) be a rational function of degreeD≥2. A well known theorem in dynamical systems says that if some iterate φn(z)=φring operatorφring operator···ring operatorφ(z) is a polynomial, then already φ2(z) is a polynomial. More generally, this is true for φ(z)set membership, variantK(z) for any fieldKprovided that φ is separable (i.e., φ′(z)≠0). However, it need not be true for inseparable φ. In this note we give a complete description, in all characteristics and with no separability assumptions, of the rational functions which have a polynomial iterate.