The Thermodynamic Formalism for the de Rham function: increment method

Journal Article

رابط المنشور على الويب:

http://iopscience.iop.org/article/10.1070/IM2012v076n03ABEH002590/meta;jsessionid=1E916AA6E92633C6FD919AA48D53088B.c4.iopscience

المجلة \ الصحيفة:

Izvestiya: Mathematics

رقم العدد:

76:3(2012), 431-445.

رقم الإصدار السنوي:

76:3(2012), 431-445.

الصفحات:

76:3(2012), 431-445.

مستخلص المنشور:

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 $\alpha(x)=\liminf_{h\to 0}\frac{\log(\vert F(x+h)-F(x)\vert)}{\log \vert h\vert}$ 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 $\int_{\mathbb{R}}\vert F(x+h)-F(x)\vert^{q}\,dx\sim \vert h\vert^{\zeta(q)}$ as $h \to 0$ with the Hausdorff dimension $d(\alpha)$ of $E^{(\alpha)}$ .