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Transfinite multifractal dimension spectrums


Author: Stanley C. Williams
Journal: Trans. Amer. Math. Soc. 348 (1996), 4043-4081
MSC (1991): Primary 28A78, 28A80, 60G42; Secondary 60G57
DOI: https://doi.org/10.1090/S0002-9947-96-01622-4
MathSciNet review: 1351496
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Abstract: The first order theory of the decomposition of measures with respect to dimension which has been developed by Kahane, Katznelson, Cutler, and others is extended through transfinite recursion to a $\omega _1$-order theory. Necessary and sufficient conditions for a finite regular Borel measure on $% [0,d]^{\omega _1}$ to be a $\omega _1$-order multispectrum for a finite Borel measure on $\mathbb {R}^d$ is given.


References [Enhancements On Off] (What's this?)

  • 1. Paul Bernays. Axiomatic Set Theory. Dover Publications, New York, 1991. MR 92a:04001.
  • 2. Donald L. Cohn. Measure Theory. Birkhäuser, Boston, 1980. MR 81k:28001.
  • 3. John B. Conway. A Course in Functional Analysis. Springer-Verlag, New York, second edition, 1990. p. 76. MR 91e:46001.
  • 4. Collen Cutler. The Hausdorff dimension distribution of finite measures in euclidean space. Canandian Journal of Mathematics, XXXVIII(6):1459--1484, 1988. MR 88b:28013.
  • 5. Joseph L. Doob. Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag, New York, 1984. MR 85k:31001.
  • 6. R. Holley and E.C. Waymire. Multifractal dimension and scaling exponents for strongly bounded random cascades. The Annals of Applied Probability, 2:819--845, 1992. MR 93k:60122.
  • 7. J.P. Kahane. Some Random Series of Functions. Cambridge University Press, New York, second edition, 1985. MR 87m:60119.
  • 8. J.P. Kahane. Multiplications aléatoires et dimension de Hausdorff. Ann. Inst. Poincaré, 23:1--12, 1987. MR 88h:60100.
  • 9. J.P. Kahane. Positive martingales and random measures. Chinese Ann. Math., 8b:1--12, 1987. MR 88j:60098.
  • 10. J.P. Kahane and Y. Katznelson. Décomposition des mesures selon la dimension. Colloq. Math., LVIII:269--279, 1990. MR 91g:28018.
  • 11. J.P. Kahane and Jaques Peyrière. Sur certaines martingales de Benoit Mandelbrot. Advances in Mathematics, 22:131--145, 1976. MR 55:4355.
  • 12. R. Daniel Mauldin. A representation theorem for the second dual of C[0,1]. Studia Mathematica, XLVI:197--200, 1973. MR 49:11231.
  • 13. G. Brown, G. Michon and Jaques Peyrière. On the multifractal analysis of measures. J. Statist. Phys. 66:775--790 (1992), MR 93c:58120.
  • 14. L. Olsen. Multifractal formalism. Adv. in Math. 116:82--196 (1995), CMP 1996:4.
  • 15. K.R. Parthasarthy. Introduction to Probability and Measure. Springer-Verlag, New York, 1978. MR 58:31322b.
  • 16. C.A. Rogers. Hausdorff Measures. Cambridge University Press, 1970. MR 43:7576.
  • 17. Claude Tricot, Jr. Two definitions of fractional dimension. Mathematical Proceedings of the Cambridge Philosophical Society, 91:57--74, 1982. MR 84d:28013.
  • 18. E.C. Waymire and S.C. Williams. Multiplicative cascades: Dimension spectra and dependence. Journal of Fourier Analysis and its Applications. 1995, Special issue, 589--609. CMP 1996:5.
  • 19. E.C. Waymire and S.C.Williams. Markov cascades. IMA Volume on Braveling Processes, ed. by K. Athroya and P. Jagers (in press).
  • 20. E.C. Waymire and S.C. Williams. A general cascade decomposition theory. Bulletin (New Series) of the American Mathematical Society, 31:216--222 (1994). MR 95a:60065.
  • 21. E.C. Waymire and S.C. Williams. A cascade decomposition theory with applications to Markov and exchangeable cascades. Transactions of the American Mathematical Society. 348:585--632 (1996). CMP 1995:10

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Additional Information

Stanley C. Williams
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900
Email: williams@sunfs.math.usu.edu

DOI: https://doi.org/10.1090/S0002-9947-96-01622-4
Keywords: Multifractals, dimension spectra, martingales, random measures
Received by editor(s): January 29, 1995
Article copyright: © Copyright 1996 American Mathematical Society

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