The Validity of the Multifractal Formalism: Results and Examples

✍ Scribed by Fathi Ben Nasr; Imen Bhouri; Yanick Heurteaux

• Book ID: 102564459
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 173 KB
• Volume: 165
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.2001.2025

✦ Synopsis

By obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math. 116, 82-196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure m for which the multifractal packing dimension function B m (q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above).