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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\begin{eqnarray*} \text{`the coarse multifractal spectra $=$ the Legendre transforms of the R\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 \text{`the R\

Keywords

28A80

Information

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

References

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