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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
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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

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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