The role of Billingsley dimensions in computing fractal dimensions on Cantorlike spaces
Author:
JoséManuel Rey
Journal:
Proc. Amer. Math. Soc. 128 (2000), 561572
MSC (1991):
Primary 28A78, 28A80; Secondary 58F11
Published electronically:
July 6, 1999
MathSciNet review:
1641089
Fulltext PDF Free Access
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Abstract: We consider a Cantorlike set as a geometric projection of a Bernoulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimensionlike indices in the probability space of a stochastic process. Under suitable regularity conditions we find closed formulae linking the Hausdorff, box and packing metric dimensions of the subsets of the Cantorlike set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of wellknown fractal spaces boils down to computing dimensions in the unit interval endowed with a suitable metric. We use these results to generalize density theorems in Cantorlike spaces. We also give some examples to illustrate the application of our results.
 [1]
Patrick
Billingsley, Hausdorff dimension in probability theory,
Illinois J. Math. 4 (1960), 187–209. MR 0131903
(24 #A1750)
 [2]
Patrick
Billingsley, Hausdorff dimension in probability theory. II,
Illinois J. Math. 5 (1961), 291–298. MR 0120339
(22 #11094)
 [3]
Tim
Bedford, Hausdorff dimension and box dimension in selfsimilar
sets, Proceedings of the Conference: Topology and Measure, V (Binz,
1987), Wissensch. Beitr., ErnstMoritzArndt Univ., Greifswald, 1988,
pp. 17–26. MR 1029553
(91a:58139)
 [4]
G.
Brown, G.
Michon, and J.
Peyrière, On the multifractal analysis of measures, J.
Statist. Phys. 66 (1992), no. 34, 775–790. MR 1151978
(93c:58120), http://dx.doi.org/10.1007/BF01055700
 [5]
Rufus
Bowen, Equilibrium states and the ergodic theory of Anosov
diffeomorphisms, Lecture Notes in Mathematics, Vol. 470,
SpringerVerlag, Berlin, 1975. MR 0442989
(56 #1364)
 [6]
Helmut
Cajar, Billingsley dimension in probability spaces, Lecture
Notes in Mathematics, vol. 892, SpringerVerlag, Berlin, 1981. MR 654147
(84a:10055)
 [7]
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
 [8]
Colleen
D. Cutler, The density theorem and Hausdorff inequality for packing
measure in general metric spaces, Illinois J. Math.
39 (1995), no. 4, 676–694. MR 1361528
(96m:28007)
 [9]
Chao
Shou Dai and S.
James Taylor, Defining fractals in a probability space,
Illinois J. Math. 38 (1994), no. 3, 480–500. MR 1269700
(95f:28011)
 [10]
Kenneth
Falconer, Fractal geometry, John Wiley & Sons Ltd.,
Chichester, 1990. Mathematical foundations and applications. MR 1102677
(92j:28008)
 [11]
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
 [12]
R.
Daniel Mauldin and Mariusz
Urbański, Dimensions and measures in infinite iterated
function systems, Proc. London Math. Soc. (3) 73
(1996), no. 1, 105–154. MR 1387085
(97c:28020), http://dx.doi.org/10.1112/plms/s373.1.105
 [13]
Pertti
Mattila, Geometry of sets and measures in Euclidean spaces,
Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge
University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
(96h:28006)
 [14]
M. MORÁN and J.M. REY, Singularity of selfsimilar measures with respect to Hausdorff measures, Trans. Amer. Math. Soc. 350 (1998), 22972310.
 [15]
M. MORÁN and J.M. REY, Geometry of selfsimilar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365386. CMP 98:01
 [16]
L.
Olsen, A multifractal formalism, Adv. Math.
116 (1995), no. 1, 82–196. MR 1361481
(97a:28006), http://dx.doi.org/10.1006/aima.1995.1066
 [17]
N. PATZSCHKE, Selfconformal multifractal measures, Adv. in Applied Math. 19 (1997), 486513.
 [18]
Y. PESIN and H. WEISS, On the dimension of deterministic and random Cantorlike sets, symbolic dynamics, and the EckmannRuelle conjecture, preprint.
 [19]
Y. PESIN and H. WEISS, A multifractal analysis of equilibrium measures for conformal expanding maps and Moranlike geometric constructions, preprint.
 [20]
Claude
Tricot Jr., Two definitions of fractional dimension, Math.
Proc. Cambridge Philos. Soc. 91 (1982), no. 1,
57–74. MR
633256 (84d:28013), http://dx.doi.org/10.1017/S0305004100059119
 [1]
 P. BILLINGSLEY, Hausdorff dimension in probability theory, Illinois J. Math. 4 (1960), 187209. MR 24:A1750
 [2]
 P. BILLINGSLEY, Hausdorff dimension in probability theory II, Illinois J. Math. 5 (1961), 291298. MR 22:11094
 [3]
 T. BEDFORD, Hausdorff dimension and box dimension in selfsimilar sets, Proc. Conf. Topology and Measure V, Binz, GDR (1987), 1726. MR 91a:58139
 [4]
 G. BROWN, G. MICHON and J. PEYRIERE, On the multifractal analysis of measures, J. Stat. Phys. 66 (1992), 775790. MR 93c:58120
 [5]
 R. BOWEN, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Maths. 470, SpringerVerlag, 1975. MR 56:1364
 [6]
 H. CAJAR, Billingsley dimension in probability spaces, Lecture Notes in Maths. 892, SpringerVerlag, 1981. MR 84a:10055
 [7]
 R. CAWLEY and R.D. MAULDIN, Multifractal decomposition of Moran fractals, Adv. Math. 92 (1992), 196236. MR 93b:58085
 [8]
 C. CUTLER, The density theorem and Hausdorff inequality for packing measures in general metric spaces, Illinois J. Math. 39 (1995), 676694.MR 96m:28007
 [9]
 C. DAI and S.J. TAYLOR, Defining fractals in a probability space, Illinois J. Math. 38 (1994), 480500.MR 95f:28011
 [10]
 K.J. FALCONER, Fractal geometry mathematical foundations and applications, Wiley, 1990. MR 92j:28008
 [11]
 J.E. HUTCHINSON, Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981), 71347. MR 82h:49026
 [12]
 R.D. MAULDIN and M. URBA\'{N}SKI, Infinite iterated function systems, Proc. London Math. Soc. 73 (1996), 105154.MR 97c:28020
 [13]
 P. MATTILA, Geometry of sets and measures in euclidean spaces, Cambridge University Press, 1995. MR 96h:28006
 [14]
 M. MORÁN and J.M. REY, Singularity of selfsimilar measures with respect to Hausdorff measures, Trans. Amer. Math. Soc. 350 (1998), 22972310.
 [15]
 M. MORÁN and J.M. REY, Geometry of selfsimilar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365386. CMP 98:01
 [16]
 L. OLSEN, A multifractal formalism, Adv. Math. 116 (1995), 82196. MR 97a:28006
 [17]
 N. PATZSCHKE, Selfconformal multifractal measures, Adv. in Applied Math. 19 (1997), 486513.
 [18]
 Y. PESIN and H. WEISS, On the dimension of deterministic and random Cantorlike sets, symbolic dynamics, and the EckmannRuelle conjecture, preprint.
 [19]
 Y. PESIN and H. WEISS, A multifractal analysis of equilibrium measures for conformal expanding maps and Moranlike geometric constructions, preprint.
 [20]
 C. TRICOT, Two definitions of fractional dimension, Math. Proc. Cambridge Phil. Soc. 91 (1982), 5774.MR 84d:28013
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Additional Information
JoséManuel Rey
Affiliation:
Centre for Nonlinear Dynamics and its Applications, University College London, Gower Street, London WC1E 6BT, United Kingdom
Address at time of publication:
Departamento de Análisis Económico, Universidad Complutense, Campus de Somosaguas, 28223 Madrid, Spain
Email:
ececo07@sis.ucm.es
DOI:
http://dx.doi.org/10.1090/S0002993999051667
PII:
S 00029939(99)051667
Received by editor(s):
October 28, 1997
Received by editor(s) in revised form:
April 9, 1998
Published electronically:
July 6, 1999
Additional Notes:
This research was partially supported by a postdoctoral grant (A.P.E.) from the Universidad Complutense de Madrid. A preliminary version of this paper was written while visiting the Mathematical Institute at the University of St Andrews. The author thanks Kenneth J. Falconer and the members of the Analysis Research Group at St Andrews for their kind hospitality during his stay. Many useful discussions with Manuel Morán stimulated the initial development of this research.
Dedicated:
This paper is dedicated to Professor Juan Carlos Simó
Communicated by:
Frederick W. Gehring
Article copyright:
© Copyright 1999 American Mathematical Society
