The thermodynamic formalism for the de Rham function: increment method (Q2901875)

Let \(F\in L^{1}({\mathbb R})\) be the unique continuous solution of the functional equation NEWLINE\[NEWLINEF(x) = F(3x)+\frac{1}{3}(F(3x-1)+F(3x+1)) + \frac{2}{3} (F(3x-2)+F(3x+2))NEWLINE\]NEWLINE satisfying condition the \(\int F(x)dx=1\). \(F\) is nowhere differentiable. Let NEWLINE\[NEWLINE \alpha(x)=\liminf_{h\to 0}\frac{\log |F(x+h)-F(x)|}{\log|h|},NEWLINE\]NEWLINE \(E^{\alpha}=\{x: \alpha(x)=\alpha\},\) and \(d(\alpha)\) is the Hausdorff dimension of \(E^{\alpha}\). The author investigates the function \(d(\alpha)\) and proves certain formulas for its evaluation.