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

Abstract EN

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.