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Exceptions to the multifractal formalism for discontinuous measures

Published online by Cambridge University Press:  01 January 1998

RUDOLF H. RIEDI  and
BENOIT B. MANDELBROT
RUDOLF H. RIEDI
Affiliation:
INRIA Rocquencourt, B.P. 105 78153 Le Chesnay Cedex, France; e-mail: [email protected]
BENOIT B. MANDELBROT
Affiliation:
Mathematics Department, Yale University, New Haven CT 06520-8283, USA; e-mail: [email protected]
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Abstract

In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.

Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Hölder exponents 0 and ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 123 , Issue 1 , January 1998 , pp. 133 - 157
Copyright
Cambridge Philosophical Society 1998

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