Abstract
We study the de Rham function: the unique continuous (nowhere differentiable) function F is an element of L-1 (R) with integral F(x) dx = 1 satisfying the functional equation F(x) = F(3x) + 1/3(F(3x - 1) + F(3x + 1)) + 2/3(F(3x - 2) + F(3x + 2)). We show that its pointwise Holder regularity alpha(x) = lim inf(h -> 0) 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 Holder 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 integral(R)|F(x + h) - F(x)|(q)dx similar to |h|(zeta(q)) as h -> 0 with the Hausdorff dimension d(alpha) of E-(alpha).