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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A dimension result arising from
the $L^{q}$-spectrum of a measure


Author: Sze-Man Ngai
Journal: Proc. Amer. Math. Soc. 125 (1997), 2943-2951
MSC (1991): Primary 28A80; Secondary 28A78
MathSciNet review: 1402878
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Abstract: We give a rigorous proof of the following heuristic result: Let $\mu $ be a Borel probability measure and let $\tau (q)$ be the $L^{q}$-spectrum of $\mu $. If $\tau (q)$ is differentiable at $q=1$, then the Hausdorff dimension and the entropy dimension of $\mu $ equal $\tau '(1)$. Our result improves significantly some recent results of a similar nature; it is also of particular interest for computing the Hausdorff and entropy dimensions of the class of self-similar measures defined by maps which do not satisfy the open set condition.


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Additional Information

Sze-Man Ngai
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
Email: smngai@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03974-9
PII: S 0002-9939(97)03974-9
Keywords: Entropy dimension, Hausdorff dimension, $L^{q}$-spectrum
Received by editor(s): February 28, 1996
Received by editor(s) in revised form: May 7, 1996
Additional Notes: Research supported by a postdoctoral fellowship of the Chinese University of Hong Kong.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society