On the derivative of iterations of the Minkowski question mark function at special points

Abstract

For the Minkowski question mark function ?(x) we consider derivative of the function fn(x) = ?(?(...? } {{ } n times (x))). Apart from obvious cases (rational numbers for example) it is non-trivial to find explicit examples of numbers x for which f ′ n (x) = 0. In this paper we present a set of irrational numbers, such that for every element x0 of this set and for any n ∈ Z+ one has f ′ n (x0) = 0.