On the derivative of the \(\alpha\)-Farey-Minkowski function (Q379748)

Denote by \(\alpha := \{A_n : n\in \mathbb{N}\}\) a countably infinite partition of the unit interval into nonempty, right-closed and left-open intervals. The author studies the family of \(\alpha\)-Farey-Minkowski maps \(\theta_\alpha\), i.e., the conjugating homeomorphisms between die \(\alpha\)-Farey map \(F_\alpha\) and the tent map \(T := \max\{0, 1- |2x-1|\}\). Two results are proven in this paper: Firstly, it is shown that for every partition \(\alpha\) which is different from the dyadic partition \(\alpha_D\), \(\theta' (x)\in \{0, \infty\}\), provided \(\theta' (x)\) exists and is finite or \(\theta' (x) = \infty\). In case \(\alpha = \alpha_D\), \(\theta_{\alpha_D} = \mathrm{id}_{[0,1]}\). As a consequence one obtains that \(\theta_{\alpha}\) is singular with respect to Lebesgue measure, and that \([0,1]\) is therefore the disjoint union of the sets \(\Theta_0 := \{x\in [0,1] : \theta'(x) = 0\}\), \(\Theta_\infty := \{x\in [0,1] : \theta'(x) = \infty\}\) and \(\Theta_\sim := \{x\in [0,1] : \text{}\theta'(x)\) does not exit