The validity of the multifractal formalism: Results and examples

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 hake 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 mu for which the multifractal packing dimension function B-n(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). (C) 2002 Elsevier Science (USA).