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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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