Multifractal Formalism for Innnite Multinomial Measures

There are strong reasons to believe that the multifractal spectrum of DLA shows anomalies which have been termed left sided ME, M]. In order to show that this is compatible with strictly multiplicative structures Mandelbrot et al. M, MEH] introduced a one parameter family of multifractal measures invariant under innnitely many linear maps on the real line. Under the assumption that the usual multifractal formalism holds, the authors showed that the multifractal spectrum of these measure is indeed left sided, i.e. increasing over the whole range ] min ; 1. Here, it is shown that the multifractal formalism for self-similar measures does indeed hold also in the innnite case, in particular that the singularity exponents (q) satisfy the usual equation P p i q i = 1 and that the spectrum f() is the Legendre transform of (q).