Inversion Formula for Continuous Multifractals

✍ Scribed by Rudolf H Riedi; Benoit B Mandelbrot

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 303 KB
• Volume: 19
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1997.0550

✦ Synopsis

In a previous paper the authors introduced the inverse measure † of a w x probability measure on 0, 1 . It was argued that the respective multifractal † Ž . Ž . spectra are linked by the ''inversion formula'' f ␣ s ␣ f 1r␣ . Here, the statements of the previous paper are put into more mathematical terms and proofs are given for the inversion formula in the case of continuous measures. Thereby, f may stand for the Hausdorff spectrum, the packing spectrum, or the coarse grained spectrum. With a closer look at the special case of self-similar measures we offer a motivation of the inversion formula as well as a discussion of possible generalizations. Doing so we find a natural extension of the scope of the notation ''self-similar'' and a failure of the usual multifractal formalism.