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The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces

Author: José-Manuel Rey
Journal: Proc. Amer. Math. Soc. 128 (2000), 561-572
MSC (1991): Primary 28A78, 28A80; Secondary 58F11
Published electronically: July 6, 1999
MathSciNet review: 1641089
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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.

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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

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