Wavelet analysis of fractal Boundaries, Part 1: Local regularity
Résumé
Let Ω be a domain of Rd. In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of Ω. Classifica- tions based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local Lp regularity of the characteristic function 1Ω, or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of Ω. 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.