All solutions to a Schröder type functional equation

We determine the solutions on various intervals in \([0,\infty [\) to the functional equation \(f(x^m)=r f(x)\) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set \({\mathcal {S}}\) of all solutions. The formulas for \(r<0\) are more complicated. An approach to \({\mathcal {S}}\) with the help of the axiom of choice is also given. A special attention is laid on solutions that are continuous on \([0,\infty [\) or on various open subintervals. We also describe solutions satisfying some asymptotic properties at the boundary of these intervals.