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

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 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 as , where and . The Knopp function itself has everywhere the same p -exponent . 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 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.