A dimension result arising from the $L^q$-spectrum of a measure
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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
- M. Arbeiter and N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5-42.
- J. C. Alexander and J. A. Yorke, Fat baker’s transformations, Ergodic Theory Dynam. Systems 4 (1984), no. 1, 1–23. MR 758890, DOI 10.1017/S0143385700002236
- J. C. Alexander and Don Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc. (2) 44 (1991), no. 1, 121–134. MR 1122974, DOI 10.1112/jlms/s2-44.1.121
- Robert Cawley and R. Daniel Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), no. 2, 196–236. MR 1155465, DOI 10.1016/0001-8708(92)90064-R
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- A.-H. Fan, Multifractal analysis of infinite products, J. Statist. Phys. 86 (1997), 1313-1336.
- J. S. Geronimo and D. P. Hardin, An exact formula for the measure dimensions associated with a class of piecewise linear maps, Constr. Approx. 5 (1989), no. 1, 89–98. Fractal approximation. MR 982726, DOI 10.1007/BF01889600
- Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Itamar Procaccia, and Boris I. Shraiman, Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A (3) 33 (1986), no. 2, 1141–1151. MR 823474, DOI 10.1103/PhysRevA.33.1141
- H. G. E. Hentschel and Itamar Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), no. 3, 435–444. MR 719636, DOI 10.1016/0167-2789(83)90235-X
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- S.P. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, preprint.
- K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math. (to appear).
- —, $L^{q}$-spectrum of the Bernoulli convolution associated with the golden ratio, preprint.
- François Ledrappier and Anna Porzio, A dimension formula for Bernoulli convolutions, J. Statist. Phys. 76 (1994), no. 5-6, 1307–1327. MR 1298104, DOI 10.1007/BF02187064
- Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, J. Statist. Phys. 86 (1997), 233-275.
- D. A. Rand, The singularity spectrum $f(\alpha )$ for cookie-cutters, Ergodic Theory Dynam. Systems 9 (1989), no. 3, 527–541. MR 1016670, DOI 10.1017/S0143385700005162
- A. Rényi, Probability theory, North-Holland Series in Applied Mathematics and Mechanics, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated by László Vekerdi. MR 0315747
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- Robert S. Strichartz, Self-similar measures and their Fourier transforms. I, Indiana Univ. Math. J. 39 (1990), no. 3, 797–817. MR 1078738, DOI 10.1512/iumj.1990.39.39038
- Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI 10.1017/s0143385700009615
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
- 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