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 localLpregularity exponents (the so-calledp-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function which is defined forx∈0, 1asFx=∑j=0∞2-αjϕ2jx, where0<α<1andϕx=distx, z. The Knopp function itself has everywhere the samep-exponentα. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute thep-exponent of the characteristic function of the domain under the graph ofFat each point(x, F(x))and show thatp-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.