Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T13:22:45.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Fractal properties of products and projections of measures in ℝd

Published online by Cambridge University Press:  24 October 2008

Xiaoyu Hu  and
S. James Taylor
Xiaoyu Hu
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
S. James Taylor
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Borel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 3 , May 1994 , pp. 527 - 544
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Besicovitch, A. S. and Moran, P. A. P.. The measure of product and cylinder sets. J. London Math. Soc. 20 (1954), 110–120.Google Scholar
[2]Cutler, C. D.. The Hausdorif dimension distribution of finite measures in Euclidean space. Can. J. Math. 38 (1986), 1459–1484.CrossRefGoogle Scholar
[3]Cutler, C. P.. Measure disintegrations with respect to σ-stable monotone indices and the pointwise representation of packing measures. Rendi del Circolo Matematico de Palermo 28 Supp (1992), 313–339.Google Scholar
[4]Falconer, K. J.. The Geometry of Fractals (Cambridge University Press, 1985).CrossRefGoogle Scholar
[5]Frostman, O.. Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Medd. Lunel. U. Math. Seminars (1935), 1–118.Google Scholar
[6]Hasse, H.. On the dimension of product measures. Mathematika 37 (1990), 316–323.CrossRefGoogle Scholar
[7]Kaufman, R.. Entropy, Dimension, and Random sets. Quart. J. Math. Oxford Set. (2), 38 (1987), 77–80.CrossRefGoogle Scholar
[8]Marstrand, J. M.. Some fundamental geometrical properties of plane sets of fractal dimension. Proc. London Math. Soc. 4 (1954), 257–302.CrossRefGoogle Scholar
[9]Rogers, C. A. and Taylor, S. J.. Additive set functions in Euclidean space. Acta Math. 101 (1959), 273–302.CrossRefGoogle Scholar
[10]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorif measure. Mathematika 8 (1962), 1–31.CrossRefGoogle Scholar
[11]Taylor, S. J.. On the connection between Hausdorff measures and generalized capacity. Math. Proc. Cambridge Phil. Soc. 57 (1961), 542–531.CrossRefGoogle Scholar
[12]Taylor, S. J.. A measure theory definition of fractals. Rendi del Gircolo Mathematico de Palermo 28 Supp (1992), 371–378.Google Scholar
[13]Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[14]Tricot, C.. Douze definitions de la densité logarithmique. Comptes Rendues 293 (1981), 549–552.Google Scholar
[15]Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Phil. Soc. 91(1981), 57–74.CrossRefGoogle Scholar