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 \[f_n(x) = \underbrace{?(?(\ldots ?}_\text{n times}(x))).\] 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 $x_0$ of this set and for any $n\in\mathbb{Z}_+$ one has $f'_n(x_0)=0$.