## 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

- 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

## 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