Advances in Applied Mathematics Multifractal Formalism for In nite 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 struc tures Mandelbrot et al M MEH introduced a one parameter family of multifractal measures invariant under in nitely 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 mea sure is indeed left sided i e increasing over the whole range min Here it is shown that the multifractal formalism for self similar mea sures does indeed hold also in the in nite case in particular that the singularity exponents q satisfy the usual equation P pi q i and that the spectrum f is the Legendre transform of q Introduction and Summary In M MEH introduced self similar multifractal measures which are constructed with an in nitely multiplicative cascade Data suggest that the measures considered in MEH show the same anomalies as observed with DLA namely the partition sum q fails to scale like q for q This be haviour is linked with the existence of in nite H older exponents and with a multifractal spectrum which is increasing over the whole range min hereafter called left sided spectrum This observation had led some authors to describe DLA as non multifractal On the other hand the in nite multi nomial measures show that left sided spectra are compatible with strictly multiplicative hence renormalizable structures This paper provides a rigorous mathematical frame for the statements made in MEH Note added in proof In nite systems of conformal contractions are studied in MU comparing packing and Hausdor measures of invariant sets The organization is as follows Following Hutchinson Hut we will start by de ning the codespace I a set of sequences i i suitable to model the in variant set of a given in nite family of contractions fwigi IIN of IR d There is a probability distribution on I such that an address picked randomly satis es in k with probability pk independent of n Via the addressing this trans lates into a probability measure on IR which is the only one satisfying the invariance X