Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



A dimension result arising from
the $L^{q}$-spectrum of a measure

Author: Sze-Man Ngai
Journal: Proc. Amer. Math. Soc. 125 (1997), 2943-2951
MSC (1991): Primary 28A80; Secondary 28A78
MathSciNet review: 1402878
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [AP] M. Arbeiter and N. Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5-42. CMP 97:01
  • [AY] J.C. Alexander and J.A. Yorke, Fat baker's transformations, Ergodic Theory Dynamical Systems. 4 (1984), 1-23. MR 86c:58090
  • [AZ] J.C. Alexander and D. Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc. 44 (1991), 121-134. MR 92g:28035
  • [CM] R. Cawley and R.D. Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), 196-236. MR 93b:58085
  • [E] P. Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974-976. MR 1:52a
  • [F] K.J. Falconer, Fractal geometry-Mathematical foundations and applications, John Wiley and Sons, New York, 1990. MR 92j:28008
  • [Fa] A.-H. Fan, Multifractal analysis of infinite products, J. Statist. Phys. 86 (1997), 1313-1336.
  • [GH] 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), 89-98. MR 90d:58076
  • [H] T.C. Halsey, M. H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986), 1141-1151. MR 87h:58125a
  • [HP] H. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica 8D (1983), 435-444. MR 85a:58064
  • [Hut] J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
  • [La] S.P. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, preprint.
  • [LN1] K.-S. Lau and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math. (to appear).
  • [LN2] -, $L^{q}$-spectrum of the Bernoulli convolution associated with the golden ratio, preprint.
  • [LP] F. Ledrappier and A. Porzio, A dimension formula for Bernoulli convolutions, J. Statist. Phys. 76 (1994), 1307-1327. MR 95i:58111
  • [PW] 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.
  • [R] D.A. Rand, The singularity spectrum $f(\alpha )$ for cookie-cutters, Ergodic Theory Dynamical Systems. 9 (1989), 527-541. MR 90k:58115
  • [Re] A. Rényi, Probability Theory, North-Holland, Amsterdam, 1970. MR 47:4296
  • [Ro] R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton, New Jersey, 1970. MR 43:445
  • [S] R.S. Strichartz, Self-similar measures and their Fourier transforms I, Indiana Univ. Math. J. 39 (1990), 797-817. MR 92k:42015
  • [Y] L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems. 2 (1982), 109-124. MR 84h:58087

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28A80, 28A78

Retrieve articles in all journals with MSC (1991): 28A80, 28A78

Additional Information

Sze-Man Ngai
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong

Keywords: Entropy dimension, Hausdorff dimension, $L^{q}$-spectrum
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
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society