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ON THE INVERSE MULTIFRACTAL FORMALISM

Published online by Cambridge University Press:  04 December 2009

L. OLSEN
L. OLSEN*
Affiliation:
Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland e-mail: lo@st-and.ac.uk
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Abstract

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Two of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Rényi dimensions. In the 1980s it was conjectured in the physics literature that for ‘good’ measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Rényi dimensions, holds, namely This result is known as the multifractal formalism and has now been verified for many classes of measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Rényi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism fails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse of the multifractal formalism, namely

Keywords

28A80
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 179 - 194
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Badii, R. and Politi, A., Complexity: Hierarchical structures and scaling in physics, Cambridge Nonlinear Science Series Vol. 6 (Cambridge University Press, Cambridge, England 1997).CrossRefGoogle Scholar
2.Beck, C. and Schlögl, F., Thermodynamics of chaotic systems: An introduction, Cambridge Nonlinear Science Series Vol. 4 (Cambridge University Press, Cambridge, England 1993).CrossRefGoogle Scholar
3.Falconer, K. J., Techniques in fractal geometry (John Wiley and Sons, Chichester 1997).Google Scholar
4.Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. and Shraiman, B. J., Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986), 1141–1151.CrossRefGoogle ScholarPubMed
5.Hentschel, H. and Procaccia, I., The infinite number of generalized dimensions of fractals and strange attractors, Physica 8D (1983), 435–444.Google Scholar
6.Lau, K.-S., Lp-spectrum and multifractal formalism, Fractal geometry and stochastics (Finsterbergen, 1994), 55–90, Prog. Probab., 37, Birkhäuser, Basel, 1995.Google Scholar
7.Olsen, L., Multifractal geometry, in Proceeding, Fractal Geometry and Stochastics II, Greifswald, Germany, August 1998. Progress in Probability, Vol. 46 (Editors Bandt, C., Graf, S. and Zähle, M.), 1–37. Birkhäuser Verlag, 2000Google Scholar
8.Olsen, L., Mixed divergence points of self-similar measures, Indiana Univ. Math. J. 52 (2003), 1343–1372.CrossRefGoogle Scholar
9.Olsen, L., Mixed generalized dimensions of self-similar measures, J. Math. Anal. Appl. 306 (2005), 516–539.CrossRefGoogle Scholar
10.Rényi, A., Some fundamental questions of information theory, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl 10 (1960), 251–282.Google Scholar
11.Rényi, A., On measures of entropy and information, in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1960 (University of California Press, Berkeley, 1961), 547–561.Google Scholar