A dimension result arising from the $L^q$-spectrum of a measure
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- by Sze-Man Ngai
- Proc. Amer. Math. Soc. 125 (1997), 2943-2951
- DOI: https://doi.org/10.1090/S0002-9939-97-03974-9
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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.References
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Bibliographic Information
- Sze-Man Ngai
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
- Email: smngai@math.cuhk.edu.hk
- 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
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2943-2951
- MSC (1991): Primary 28A80; Secondary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-97-03974-9
- MathSciNet review: 1402878