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

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.