A dimension result arising from

the -spectrum of a measure

Author:
Sze-Man Ngai

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a rigorous proof of the following heuristic result: Let be a Borel probability measure and let be the -spectrum of . If is differentiable at , then the Hausdorff dimension and the entropy dimension of equal . 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.

**[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]**-,*-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 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**

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

Email:
smngai@math.cuhk.edu.hk

DOI:
https://doi.org/10.1090/S0002-9939-97-03974-9

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