Derivative of the iterations of Minkowski question mark function in special points

Abstract

For the Minkowski question mark function $?(x)$ we consider derivative of the function $f_n(x)=\underset{\text{n times}}{\underbrace{?(?(...?}}(x)))$. Apart from obvious cases (rational numbers for example) it is non-trivial to find explicit examples of numbers $x$ for which $f_n^{\prime }(x)=0$. 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 $f_n^{\prime }(x_0)=0$.

11 pages