Analysis of a fractal boundary: the graph of the Knopp function. (English) Zbl 1470.28006

Ben Slimane, Mourad; Mélot, Clothilde

Abstr. Appl. Anal. 2015, Article ID 587347, 14 p. (2015).

Summary: 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 \(x \in \left[0, 1\right]\) as \(F \left(x\right) = \sum_{j = 0}^\infty 2^{- \alpha j} \phi \left(2^j x\right)\), where \(0 < \alpha < 1\) and \(\phi \left(x\right) = \text{dist} \left(x, \mathbb{Z}\right)\). 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 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.

MSC:

28A80

Fractals

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