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Directional Thermodynamic Formalism

1
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2
Ecole Supérieure des Sciences et de la Technologie de Hammam Sousse, Université de Sousse, Sousse 4011, Tunisia
3
Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(6), 825; https://doi.org/10.3390/sym11060825
Received: 5 April 2019 / Revised: 2 June 2019 / Accepted: 4 June 2019 / Published: 21 June 2019
(This article belongs to the Special Issue Symmetry and Complexity 2019)
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PDF [481 KB, uploaded 26 June 2019]

Abstract

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets. View Full-Text
Keywords: directional hölder regularity; anisotropic hölder regularity; directional scaling function; anisotropic scaling function; directional multifractal formalism; wavelet bases; sierpinski cascade functions; fractional brownian sheets directional hölder regularity; anisotropic hölder regularity; directional scaling function; anisotropic scaling function; directional multifractal formalism; wavelet bases; sierpinski cascade functions; fractional brownian sheets
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ben Slimane, M.; Ben Abid, M.; Ben Omrane, I.; Halouani, B. Directional Thermodynamic Formalism. Symmetry 2019, 11, 825.

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