The Hausdorff and Packing Dimensions of Some Sets Related to Sierpiński Carpets

Ole A. Nielsen

Type: Article

Publication Date: 1999-10-01

Citations: 25

DOI: https://doi.org/10.4153/cjm-1999-047-4

Abstract

Abstract The Sierpiński carpets first considered by C.McMullen and later studied by Y. Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i) calculates theHausdorff, box (or Minkowski), and packing dimensions of themodified Sierpiński carpets and (ii) shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.

Locations

Canadian Journal of Mathematics - View - PDF

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Works Cited by This (6)

Action	Title	Year	Authors
+	The packing measure of self-affine carpets	1994	Yuval Peres
+ PDF Chat	Fractal Geometry: Mathematical Foundations and Applications.	1990	K. J. Falconer
+ PDF Chat	Packing measure, and its evaluation for a Brownian path	1985	S. J. Taylor Claude Tricot
+	The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure	1994	Yuval Peres
+ PDF Chat	The Hausdorff dimension of general Sierpiński carpets	1984	Curt McMullen
+	Geometry of Sets and Measures in Euclidean Spaces	1995	Pertti Mattila