Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-14T04:50:44.094Z Has data issue: false hasContentIssue false
  • English
  • Français

Local dimension and regular points

Published online by Cambridge University Press:  24 October 2008

Martin T. Barlow  and
S. James Taylor
Martin T. Barlow
Affiliation:
Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB
S. James Taylor
Affiliation:
Department of Mathematics, Maths-Astronomy Building, University of Virginia, Charlottesville, VA 22903, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

The Hausdorff–Besicovitch dimension of a set A ⊆ ℝd, denoted dim (A), relates to the structure of A in the neighbourhood of its thickest point. If A is irregular then one may wish for an index which will describe the size of A in different places. The obvious definition of the local dimension of A near x is

and it is not hard to verify that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 2 , September 1992 , pp. 361 - 368
Copyright
Copyright © Cambridge Philosophical Society 1992

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]Barlow, M. T. and Taylor, S. J.. Defining fractal subsets of ℤd. Proc. London Math. Soc. (3) 64 (1992), 125–152.CrossRefGoogle Scholar
[2]Davies, R. O.. A property of Hausdorff measures. Proc. Cambridge Philos. Soc. 52 (1956), 30–32.CrossRefGoogle Scholar
[3]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[4]Frostman, O.. Potential d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Medd. Lunds. Math. Sem. 3 (1935), 1–118.Google Scholar
[5]Landhof, N. S.. Foundations of Modern Potential Theory (Springer-Verlag, 1972).CrossRefGoogle Scholar
[6]Rogers, C. A.. Hausdorff Measures (Cambridge University Press, 1970).Google Scholar
[7]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961), 1–31.CrossRefGoogle Scholar
[8]Rogers, C. A.. Sets non-σ-finite for Hausdorff measure. Mathematika 9 (1962), 95–103.CrossRefGoogle Scholar
[9]Sion, M. and Sjerve, D.. Approximation properties of measures generated by continuous set functions. Mathematika 9 (1962) 145–156.CrossRefGoogle Scholar
[10]Taylor, S. J.. On the connection between Hausdorff measures and generalised capacities. Proc. Cambridge Philos. Soc. 57 (1961), 524–531.CrossRefGoogle Scholar