Analysis of a Fractal Boundary: The Graph of the Knopp Function

Abstract

A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local $L^p$ regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined for $x\in \left[0, 1\right]$ as $F\left(x\right)={\sum }_{j=0}^{\infty }{2}^{-\alpha j}\varphi \left(2^{j}x\right)$, where and $\varphi \left(x\right)=\text{dist}\left(x, \mathbb{z}\right)$. The Knopp function itself has everywhere the same p-exponent $\alpha$. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of the domain under the graph of F at each point $(x, F(x))$ and show that p-exponents, weak and strong accessibility exponents, change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents.