ON THE INVERSE MULTIFRACTAL FORMALISM

Two of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Renyi dimensions. In the 1980s it vas conjectured in the physics literature that for 'good' Measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Renyi dimensions, holds, namely 'the coarse multifractal spectra = the Legendre transforms of the Renyi dimensions'. This result is known as the multifractal formalism and has now been verified for mail), classes Of Measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Renyi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism rails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse or the multifractal formalism, namely 'the Renyi dimensions = the Legendre transforms of the coarse multifractal spectra'.