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Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

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Abstract

Let be a domain of . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of . 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 or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of . 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.

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Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Stéphane Jaffard

  2. LATP, CMI, Université de Provence, 39 rue F. Joliot-Curie, 13453, Marseille Cedex 13, France

    Clothilde Mélot

Authors
  1. Stéphane Jaffard
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  2. Clothilde Mélot
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Correspondence to Stéphane Jaffard.

Additional information

Communicated by M. B. Ruskai

The first author is supported by the Institut Universitaire de France.

This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII, France) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.

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Jaffard, S., Mélot, C. Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents. Commun. Math. Phys. 258, 513–539 (2005). https://doi.org/10.1007/s00220-005-1354-1

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