Hausdorff dimension of extremely slow minimal dynamical systems and Hölder preserving differentiable extensions

We study differentiable functions $f$ from compact perfect subsets $\mathfrak{C}$ of ${\mathbb R}$ onto $\mathfrak{C}$ with vanishing derivative, that is, with $f'(x)=0$ for every $x\in\mathfrak{C}$. We show that the domain of such a function can have Hausdorff dimension $d$ for any $d\in[0,1)$ and that it can be extended to a differentiable function $F\colon {\mathbb R}\to{\mathbb R}$ such that $F$ is $\alpha$-Hölder for every $\alpha\in(0,1)$. This last part is deduced from a novel generalization of Jarník's differentiable extension theorem stating that every differentiable map $f\colon P\to{\mathbb R}$, where $P\subset {\mathbb R}$ is compact, admits a differentiable extension $F\colon{\mathbb R}\to{\mathbb R}$ which preserves Hölder continuity of $f$.