Multifractal Formalism for Infinite Multinomial Measures

✍ Scribed by R.H. Riedi; B.B. Mandelbrot

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 654 KB
• Volume: 16
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1995.1007

✦ Synopsis

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 and co-workers 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, they showed that the multifractal spectrum of these measures is indeed left-sided, i.e., increasing over the whole (\alpha) range (] \alpha_{\min }, x[). 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 (\tau(q)) satisfy the usual equation (\sum p_{i}^{q} \lambda_{i}^{\top}=1) and that the spectrum (f(\alpha)) is the Legendre transform of (\tau(q)). 1995 Academic Press, Inc.