Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

Let [inline-graphic not available: see fulltext] be a domain of [inline-graphic not available: see fulltext]. In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of [inline-graphic not available: see fulltext]. Classifications 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 [inline-graphic not available: see fulltext] or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of [inline-graphic not available: see fulltext]. 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.