On the derivative of the α-Farey-Minkowski function

Abstract

In this paper we study the family of $α$-Farey-Minkowski functions $θ_α$, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $α$-Farey systems and the tent map. We first show that each function $θ_α$ 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: $Θ_0:={x\in\U:θ_α'(x)=0}, Θ_\infty:={x\in\U:θ_α'(x)=\infty} and Θ_\sim:=\U\setminus(Θ_0\cupΘ_\infty)$. The main result is that [\dim_{\mathrm{H}}(Θ_\infty)=\dim_{\mathrm{H}}(Θ_\sim)=σ_α(\log2)<\dim_{\mathrm{H}}(Θ_0)=1,] where $σ_α(\log2)$ is the Hausdorff dimension of the level set ${x\in \U:Λ(F_α, x)=s}$, where $Λ(F_α, x)$ is the Lyapunov exponent of the map $F_α$ at the point $x$. The proof of the theorem employs the multifractal formalism for $α$-Farey systems.