A dimension result arising from the spectrum of a measure
Author:
SzeMan Ngai
Journal:
Proc. Amer. Math. Soc. 125 (1997), 29432951
MSC (1991):
Primary 28A80; Secondary 28A78
MathSciNet review:
1402878
Fulltext PDF Free Access
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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 selfsimilar measures defined by maps which do not satisfy the open set condition.
 [AP]
M. Arbeiter and N. Patzschke, Random selfsimilar multifractals, Math. Nachr. 181 (1996), 542. CMP 97:01
 [AY]
J.
C. Alexander and J.
A. Yorke, Fat baker’s transformations, Ergodic Theory
Dynam. Systems 4 (1984), no. 1, 1–23. MR 758890
(86c:58090), http://dx.doi.org/10.1017/S0143385700002236
 [AZ]
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
(92g:28035), http://dx.doi.org/10.1112/jlms/s244.1.121
 [CM]
Robert
Cawley and R.
Daniel Mauldin, Multifractal decompositions of Moran fractals,
Adv. Math. 92 (1992), no. 2, 196–236. MR 1155465
(93b:58085), http://dx.doi.org/10.1016/00018708(92)90064R
 [E]
Paul
Erdös, On a family of symmetric Bernoulli convolutions,
Amer. J. Math. 61 (1939), 974–976. MR 0000311
(1,52a)
 [F]
Kenneth
Falconer, Fractal geometry, John Wiley & Sons, Ltd.,
Chichester, 1990. Mathematical foundations and applications. MR 1102677
(92j:28008)
 [Fa]
A.H. Fan, Multifractal analysis of infinite products, J. Statist. Phys. 86 (1997), 13131336.
 [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), no. 1, 89–98. Fractal approximation.
MR 982726
(90d:58076), http://dx.doi.org/10.1007/BF01889600
 [H]
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
(87h:58125a), http://dx.doi.org/10.1103/PhysRevA.33.1141
 [HP]
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
(85a:58064), http://dx.doi.org/10.1016/01672789(83)90235X
 [Hut]
John
E. Hutchinson, Fractals and selfsimilarity, Indiana Univ.
Math. J. 30 (1981), no. 5, 713–747. MR 625600
(82h:49026), http://dx.doi.org/10.1512/iumj.1981.30.30055
 [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]
François
Ledrappier and Anna
Porzio, A dimension formula for Bernoulli convolutions, J.
Statist. Phys. 76 (1994), no. 56, 1307–1327.
MR
1298104 (95i:58111), http://dx.doi.org/10.1007/BF02187064
 [PW]
Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moranlike geometric constructions, J. Statist. Phys. 86 (1997), 233275.
 [R]
D.
A. Rand, The singularity spectrum 𝑓(𝛼) for
cookiecutters, Ergodic Theory Dynam. Systems 9
(1989), no. 3, 527–541. MR 1016670
(90k:58115), http://dx.doi.org/10.1017/S0143385700005162
 [Re]
A.
Rényi, Probability theory, NorthHolland Publishing
Co., AmsterdamLondon; American Elsevier Publishing Co., Inc., New York,
1970. Translated by László Vekerdi; NorthHolland Series in
Applied Mathematics and Mechanics, Vol. 10. MR 0315747
(47 #4296)
 [Ro]
R.
Tyrrell Rockafellar, Convex analysis, Princeton Mathematical
Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
(43 #445)
 [S]
Robert
S. Strichartz, Selfsimilar measures and their Fourier transforms.
I, Indiana Univ. Math. J. 39 (1990), no. 3,
797–817. MR 1078738
(92k:42015), http://dx.doi.org/10.1512/iumj.1990.39.39038
 [Y]
Lai
Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic
Theory Dynamical Systems 2 (1982), no. 1,
109–124. MR
684248 (84h:58087)
 [AP]
 M. Arbeiter and N. Patzschke, Random selfsimilar multifractals, Math. Nachr. 181 (1996), 542. CMP 97:01
 [AY]
 J.C. Alexander and J.A. Yorke, Fat baker's transformations, Ergodic Theory Dynamical Systems. 4 (1984), 123. 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), 121134. MR 92g:28035
 [CM]
 R. Cawley and R.D. Mauldin, Multifractal decompositions of Moran fractals, Adv. Math. 92 (1992), 196236. MR 93b:58085
 [E]
 P. Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974976. MR 1:52a
 [F]
 K.J. Falconer, Fractal geometryMathematical 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), 13131336.
 [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), 8998. 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), 11411151. MR 87h:58125a
 [HP]
 H. Hentschel and I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica 8D (1983), 435444. MR 85a:58064
 [Hut]
 J.E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713747. 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), 13071327. MR 95i:58111
 [PW]
 Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moranlike geometric constructions, J. Statist. Phys. 86 (1997), 233275.
 [R]
 D.A. Rand, The singularity spectrum for cookiecutters, Ergodic Theory Dynamical Systems. 9 (1989), 527541. MR 90k:58115
 [Re]
 A. Rényi, Probability Theory, NorthHolland, 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, Selfsimilar measures and their Fourier transforms I, Indiana Univ. Math. J. 39 (1990), 797817. MR 92k:42015
 [Y]
 L.S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems. 2 (1982), 109124. MR 84h:58087
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Additional Information
SzeMan Ngai
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
Email:
smngai@math.cuhk.edu.hk
DOI:
http://dx.doi.org/10.1090/S0002993997039749
PII:
S 00029939(97)039749
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
