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Wavelet analysis of fractal boundaries. I: Local exponents. (English) Zbl 1080.28005

Summary: Let \(\Omega\) be a domain of \(\mathbb R^d\). In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of \(\Omega\). Classifications based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local \(L^{p}\) regularity of the characteristic function \(l_Q\), or on its wavelet coefficients. Part II [S. Jaffard and C. Mélot, Commun. Math. Phys. 258, No. 3, 541–565 (2005; Zbl 1081.28007)] deals with the global analysis of the boundary of \(\Omega\). We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.

MSC:

28A78 Hausdorff and packing measures
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
82C99 Time-dependent statistical mechanics (dynamic and nonequilibrium)

Citations:

Zbl 1081.28007
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