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

TL;DR

This paper investigates the derivatives of iterated Minkowski question mark functions at special irrational points, identifying a set where all derivatives of all iterates are zero, revealing new properties of this fractal function.

Contribution

It introduces a specific set of irrational numbers where all derivatives of all iterates of the Minkowski question mark function vanish, providing new insights into its differentiability properties.

Findings

Identified a set of irrational numbers with zero derivatives for all iterates

Demonstrated that derivatives at rational points are trivial or non-trivial

Extended understanding of the differentiability structure of the Minkowski question mark function

Abstract

For the Minkowski question mark function $?(x)$ we consider derivative of the function $f_n(x) = \underbrace{?(?(...?}_\text{n times}(x)))$. 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 $f'_n(x_0)=0$.