The thermodynamic formalism for the de~Rham function: increment method

M. Ben Slimane

Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 431-445

Résumé

We study the de Rham function: the unique continuous (nowhere differentiable) function $F \in L^1(\mathbb{R})$ with $\int F(x)\,dx=1$ satisfying the functional equation $F(x)=F(3x)+\frac{1}{3}\bigl(F(3x-1)+F(3x+1) \bigr)+\frac{2}{3}\bigl(F(3x-2)+F(3x+2)\bigr)$. We show that its pointwise Hölder regularity $\alpha(x)=\liminf_{h\to 0}\frac{\log(|F(x+h)-F(x)|)}{\log |h|}$ differs widely from point to point, and the values of $\alpha(x)$ fill an interval parametrizing the fractal sets $E^{(\alpha)}$, where $E^{(\alpha)}$ is the set of points $x$ with Hölder exponent $\alpha(x)=\alpha$. We also prove that the thermodynamic formalism (increment method) holds for the de Rham function: we have a heuristic formula $d(\alpha)=\inf_{q >0}(\alpha q-\zeta(q)+1)$ relating the order of decay of $\int_{\mathbb{R}}|F(x+h)-F(x)|^{q}\,dx \sim |h|^{\zeta(q)}$ as $h \to 0$ with the Hausdorff dimension $d(\alpha)$ of $E^{(\alpha)}$.

Détail

Keywords: Hölder regularity, thermodynamic formalism.
Mots-clés : Hausdorff dimension, increments