Analysis of a fractal boundary: the graph of the Knopp function

TL;DR

This paper applies a novel local regularity analysis method to the graph of the Knopp fractal function, revealing pointwise variations in accessibility and regularity exponents and characterizing local extrema.

Contribution

It demonstrates the use of accessibility and p-exponents to analyze the local behavior of the Knopp function's graph, highlighting variations at different points.

Findings

p-exponents vary across the graph, indicating local differences in regularity.

Accessibility exponents change at local maxima and minima.

The method characterizes local extrema based on exponent values.

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.