A Self-Affine Property of Evolutional Type Appearing in a Hamilton–Jacobi Flow Starting from the Takagi Function

Abstract

In this paper, we study a Hamilton–Jacobi flow ${\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}}_{\mathit{t}>0}$ starting from the Takagi function τ. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on $\mathbb{R}$. As the first result of this paper, we derive an explicit representation of $\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}$. It turns out that ${\mathit{H}}_{\mathit{t}}\mathit{\tau }$ is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that $\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\}$ has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.