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 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. 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 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.
Keywords
Cite
@article{arxiv.1407.6219,
title = {Analysis of a fractal boundary: the graph of the Knopp function},
author = {Mourad Ben Slimane and Clothilde Melot},
journal= {arXiv preprint arXiv:1407.6219},
year = {2014}
}