Analysis of a Fractal Boundary: The Graph of the Knopp Function - Mourad Ben Slimane; Clothilde Mélot (Abstract and Applied Analysis)
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 Lp 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∈0, 1 as Fx=∑j=0∞2-αjϕ2jx, where 0<α<1 and ϕx=distx, z. The Knopp function itself has everywhere the same p-exponent α. 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.