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 $L^p$ 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 $x\in[0,1]$ as
$F(x) = \sum_{j=0}^\infty 2^{-\alpha j} \phi(2^{j}x)$
where $0<\alpha <1$ and $\phi(x) = dist(x,\mathbb{Z})$. The Knopp function itself has everywhere the
same $p$-exponent $\alpha$. 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.
