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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces
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by José-Manuel Rey PDF
Proc. Amer. Math. Soc. 128 (2000), 561-572 Request permission

Abstract:

We consider a Cantor-like set as a geometric projection of a Ber- noulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimension-like 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 Cantor–like set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of well-known 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 Cantor–like spaces. We also give some examples to illustrate the application of our results.
References
  • Patrick Billingsley, Hausdorff dimension in probability theory, Illinois J. Math. 4 (1960), 187–209. MR 131903
  • Patrick Billingsley, Hausdorff dimension in probability theory. II, Illinois J. Math. 5 (1961), 291–298. MR 120339
  • Tim Bedford, Hausdorff dimension and box dimension in self-similar sets, Proceedings of the Conference: Topology and Measure, V (Binz, 1987) Wissensch. Beitr., Ernst-Moritz-Arndt Univ., Greifswald, 1988, pp. 17–26. MR 1029553
  • G. Brown, G. Michon, and J. Peyrière, On the multifractal analysis of measures, J. Statist. Phys. 66 (1992), no. 3-4, 775–790. MR 1151978, DOI 10.1007/BF01055700
  • Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
  • Helmut Cajar, Billingsley dimension in probability spaces, Lecture Notes in Mathematics, vol. 892, Springer-Verlag, Berlin-New York, 1981. MR 654147
  • 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
  • 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
  • Chao Shou Dai and S. James Taylor, Defining fractals in a probability space, Illinois J. Math. 38 (1994), no. 3, 480–500. MR 1269700
  • Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
  • 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
  • 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, DOI 10.1112/plms/s3-73.1.105
  • 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, DOI 10.1017/CBO9780511623813
  • M. Morán and J.-M. Rey, Singularity of self–similar measures with respect to Hausdorff measures, Trans. Amer. Math. Soc. 350 (1998), 2297–2310.
  • M. Morán and J.-M. Rey, Geometry of self–similar measures, Ann. Acad. Sci. Fenn. Mathematica 22 (1997), 365-386.
  • L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), no. 1, 82–196. MR 1361481, DOI 10.1006/aima.1995.1066
  • N. Patzschke, Self–conformal multifractal measures, Adv. in Applied Math. 19 (1997), 486–513.
  • Y. Pesin and H. Weiss, On the dimension of deterministic and random Cantor–like sets, symbolic dynamics, and the Eckmann–Ruelle conjecture, preprint.
  • Y. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions, preprint.
  • Claude Tricot Jr., Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 57–74. MR 633256, DOI 10.1017/S0305004100059119
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Additional Information
  • José-Manuel 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
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 561-572
  • MSC (1991): Primary 28A78, 28A80; Secondary 58F11
  • DOI: https://doi.org/10.1090/S0002-9939-99-05166-7
  • MathSciNet review: 1641089