Multifractal Formalism for Infinite Multinomial Measures
Riedi, Rudolf H.
There are strong reasons to believe that the multifractal spectrum of DLA shows anomalies which have been termed left sided. In order to show that this is compatible with strictly multiplicative structures Mandelbrot et al. introduced a one parameter family of multifractal measures invariant under infinitely 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. they possess arbitrarily large HÃ¶lder exponents and the spectrum is increasing over the whole range of these values. Here, it is shown that the multifractal formalism for self-similar measures does indeed hold also in the infinite case, in particular that the singularity exponents D(q) satisfy the usual equation of self-similar measures and that the multifractal spectrum f(a) is the Legendre transform of (q-1)D(q).