On the derivative of the \alpha-Farey-Minkowski function

TL;DR

This paper investigates the derivative properties of the lpha-Farey-Minkowski functions, revealing their singularity, fractal structure, and the Hausdorff dimensions of sets where the derivative is zero, infinite, or undefined.

Contribution

It introduces a detailed multifractal analysis of lpha-Farey-Minkowski functions, establishing the Hausdorff dimensions of key derivative-related sets.

Findings

lpha-Farey-Minkowski functions are singular w.r.t. Lebesgue measure.

The sets where the derivative is zero or infinite have Hausdorff dimension lpha(\u03bb2).

The set where the derivative is neither zero nor infinite has Hausdorff dimension lpha(\u03bb2).

Abstract

In this paper we study the family of $\alpha$-Farey-Minkowski functions $\theta_\alpha$, for an arbitrary countable partition $\alpha$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $\alpha$-Farey systems and the tent map. We first show that each function $\theta_\alpha$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $\Theta_0:={x\in\U:\theta_\alpha'(x)=0}, \Theta_\infty:={x\in\U:\theta_\alpha'(x)=\infty} and \Theta_\sim:=\U\setminus(\Theta_0\cup\Theta_\infty)$. The main result is that [\dim_{\mathrm{H}}(\Theta_\infty)=\dim_{\mathrm{H}}(\Theta_\sim)=\sigma_\alpha(\log2)<\dim_{\mathrm{H}}(\Theta_0)=1,] where $\sigma_\alpha(\log2)$ is the Hausdorff dimension of the level set ${x\in \U:\Lambda(F_\alpha, x)=s}$, where $\Lambda(F_\alpha, x)$ is the Lyapunov exponent of the map $F_\alpha$ at the point $x$. The proof of the theorem employs the multifractal formalism for $\alpha$-Farey systems.