Abstract:
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 $E^{(\alpha)}$.
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