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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

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