The validity of the multifractal formalism : Results and examples
Résumé
By obtening a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developped by Olsen (1995, Adv. Math.). 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 and then give 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 $\mu$ for which the multifractal packing dimension function $B_\mu(q)$ 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 in (cited above).