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Baire generic results for the anisotropic multifractal formalism

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Abstract

Ben Slimane (Math Proc Camb Philos Soc 124:329–363, 1998) has constructed specific anisotropic selfsimilar functions as counter-examples for the isotropic multifractal formalism. An anisotropic multifractal formalism has been formulated and its validity for anisotropic selfsimilar functions has been proved. In this paper, using Triebel anisotropic wavelet decompositions, we first obtain lower bounds of the anisotropic scaling function and upper bounds of the u-spectrum of singularities valid for all functions. We then investigate the generic validity, in the sense of Baire’s categories, of the anisotropic formalism in some anisotropic functional spaces. We thus extend in the anisotropic setting some results of Jaffard (J Math Pure Appl 79:525–552, 2000, Ann Appl Probab 10:313–329, 2000) and Jaffard and Meyer (Memoirs of the American Mathematical Society, vol. 123, 1996).

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Acknowledgments

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding Research Group no. RG-1435-063. Mourad Ben Slimane is thankful to Stéphane Jaffard for stimulating discussions.

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Authors and Affiliations

  1. King Saud University, Department of Mathematics, College of Science, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

    Mourad Ben Slimane

  2. Laboratoire de Recherche Équations aux Dérivées Partielles et Applications, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Hnia Ben Braiek

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  1. Mourad Ben Slimane
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Ben Slimane, M., Ben Braiek, H. Baire generic results for the anisotropic multifractal formalism. Rev Mat Complut 29, 127–167 (2016). https://doi.org/10.1007/s13163-015-0185-7

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