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

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  • 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 [0,1] : \theta_\alpha'(x)=0\}, \Theta_{\infty} : = \{ x \in [0,1] : \theta_\alpha'(x)=\infty \} $ and $\Theta_\sim : = \{ x \in [0,1] : \theta_\alpha'(x)\ does\ not\ exist \} $. 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 $\sigma_\alpha(\log2)$ denotes the Hausdorff dimension of the level set $\{x\in [0,1]:\Lambda(F_\alpha, x)=\log2\}$ and $\Lambda(F_\alpha, x)$ 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.
    Mathematics Subject Classification: Primary: 26A30; Secondary: 37C45.


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