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Two definitions of fractional dimension

Published online by Cambridge University Press:  24 October 2008

Claude Tricot Jr
Claude Tricot Jr
Affiliation:
University of Liverpool
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Extract

The main properties of the Hausdorff dimension, here denoted by dim, are

In ℝp, in variance under a group Н of homeomorphisms: ∀HεH, dim О H = dim. The definition of H, introduced in (15) and (16), is recalled in § 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 1 , January 1982 , pp. 57 - 74
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Beardon, A. F.The generalized capacity of Cantor sets, Quart. J. Math. Oxford (2) 19 (1968), 301–304.CrossRefGoogle Scholar
(2)Besicovitch, A. S. and Moban, P. A. P.The measure of product and cylinder sets. J. London Math. Soc. 20 (1945), 110–120.Google Scholar
(3)Billingsley, P.Hausdorff dimension in probability theory. Illinois J. Math. 4 (1960), 187–209.CrossRefGoogle Scholar
(4)Davies, R. O.Subsets of finite measure in analytic sets. Indagat. Math. 14 (1952), 488–489.CrossRefGoogle Scholar
(5)Eggileston, H. G.A correction to a paper on the dimension of cartesian product sets. Proc. Camb. Phil. Soc. 49 (1953), 437–440.CrossRefGoogle Scholar
(6)Hawkes, J.Hausdorff measure, entropy, and the independence of small sets. Proc. London Math. Soc. (3) 28 (1974), 700–724.CrossRefGoogle Scholar
(7)Kahane, J. P. et Salem, R.Ensembles parfaits et series trigonométriques. Paris, Hermann, 1963.Google Scholar
(8)Kaufman, R.On Hausdorff dimension of projections. Mathematika 15 (1968), 153–155.CrossRefGoogle Scholar
(9)Kolmogorov, A. N. and Tihomirov, V. M.e-Entropy and e-capacity of sets in functional spaces. Amer. Math. Soc. Transl. 17 (1961), 277–364.Google Scholar
(10)Larman, D. G.On Hausdorff measure in finite-dimensional compact metric spaces. Proc. London Math. Soc. (3), 17 (1967), 193–206.CrossRefGoogle Scholar
(11)Marstrand, J. M.The dimension of the cartesian product sets. Proc. Camb. Phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
(12)Rogers, C. A.Hausdorff measures. Cambridge University Press, 1970.Google Scholar
(13)Rogers, C. A. and Taylor, S. J.Additive set functions in euclidean space. Ada Math. 101 (1959), 273–302.Google Scholar
(14)Tricot, C. Sur la notion de densité. Cahiers du Dep. d'Econométrie de l'Université de Genève (1973).Google Scholar
(15)Tricot, C. JrSur la classification des ensembles boré liens de measure de Lebesgue nulle. Genève, Imprimerie Nationale, 1980.Google Scholar
(16)Tricot, C. JrRarefaction indices. Mathematika 27 (1980), 46–57.CrossRefGoogle Scholar
(17)Wegmann, H.Die Hausdorff-Dimension von kartesischen Produktmengen in metrischen Räumen. J. Reine Angew. Math. 234 (1969), 163–171.Google Scholar