Multifractal Structure of Noncompactly Supported Measures

Authors

Sze-Man Ngai, Georgia Southern UniversityFollow

Document Type

Article

Publication Date

9-2008

Publication Title

Fractals

DOI

10.1142/S0218348X0800396X

ISSN

1793-6543

Abstract

Many important definitions in the theory of multifractal measures on ℝ^d, such as the L^q-spectrum, L^∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported or infinite measures. We propose definitions that extend the original definitions to positive Borel measures on ℝ^d which are finite on bounded sets, and recover many important results that hold for compactly supported finite measures. In particular, we prove that if the L^q-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.

Recommended Citation

Ngai, Sze-Man. 2008. "Multifractal Structure of Noncompactly Supported Measures." Fractals, 16 (3): 209-226. doi: 10.1142/S0218348X0800396X
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/120