Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T14:00:36.031Z Has data issue: false hasContentIssue false
  • English
  • Français

The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure

Published online by Cambridge University Press:  24 October 2008

Yuval Peres
Yuval Peres
Affiliation:
Yale University, Department of Mathematics
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the self-affine sets considered by McMullen in [11] and by Bedford in [1] have infinite Hausdorff measure in their dimension, except in the (rare) cases where the Hausdorff dimension coincides with the Minkowski (≡ box) dimension. More precisely, the Hausdorff measure of such a self-affine set K is infinite in the gauge

(where γ is the Hausdorff dimension of K, and c > 0 is small). The Hausdorff measure of K becomes zero if 2 is replaced by any smaller number in the formula for the gauge ø.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 513 - 526
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]Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similar sets. Ph.D. Thesis (1984), University of Warwick.Google Scholar
[2]Bedford, T.. On Weierstrass-like functions and random recurrent sets. Math. Proc. Cambridge Phil. Soc. 106 (1989), 325–342.CrossRefGoogle Scholar
[3]Billingsley, P.. Ergodic Theory and Information (Wiley, 1965).Google Scholar
[4]Bowen, R.. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470 (Springer, 1974).Google Scholar
[5]Gatzouras, D. and Lalley, S.. Hausdorif and box dimensions of certain self-affine fractals. Indiana University Math. J. 41 (1992), 533–568.Google Scholar
[6]Hoeffding, W.. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, no. 302 (1963), 13–30.CrossRefGoogle Scholar
[7]Hutchinson, J. E.. Fractals and self-similarity. Indiana University Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[8]Kenyon, R. and Peres, Y.. Hausdorif dimensions of affine-invariant sets, preprint.Google Scholar
[9]Kono, N.. On self-affine functions. Japan J. Appl. Math. 3 (1986), 259–269.CrossRefGoogle Scholar
[10]Loève, M.. Probability Theory (Van Nostrand, 1960).Google Scholar
[11]McMullen, C.. The Hausdorif dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[12]Peres, Y.. The packing measure of self-affine carpets, to appear in Math. Proc. Cambridge Phil.Soc.Google Scholar
[13]Rogers, C. A. and Taylor, S. J.. Functions continuous and singular with respect to a Hausdorif measure. Mathematika 8 (1961), 1–31.CrossRefGoogle Scholar
[14]Taylor, S. J.. The measure theory of random fractals. Math. Proc. Cambridge Phil. Soc. 100 (1986), 383–406.CrossRefGoogle Scholar
[15]Urbanski, M.. The Hausdorif dimension of the graphs of continuous self-affine functions. Proc. Amer. Math. Soc. 108 (1990), 921–930.CrossRefGoogle Scholar