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The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. (English) Zbl 0811.28005
Summary: We show that the self-affine sets considered by C. McMullen [Nagoya Math. J. 96, 1-9 (1984; Zbl 0539.28003)] and by T. Bedford [Ph. D. Thesis (1984)] have infinite Hausdorff measure in their dimension, except in the (rare) cases where the Hausdorff dimension coincides with the Minkowski (\(\equiv\) box) dimension. More precisely, the Hausdorff measure of such a self-affine set \(K\) is infinite in the gauge \[ \phi(t)= t^ \gamma\exp\left[{- c|\log t|\over (\log| \log t|)^ 2}\right], \] (where \(\gamma\) 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 \(\phi\).

28A80 Fractals
28A78 Hausdorff and packing measures
Full Text: DOI
[1] Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms 470 (1974) · Zbl 0308.28010
[2] Billingsley, Ergodic Theory and Information (1965)
[3] Bedford, Math. Proc. Cambridge Phil. Soc 106 pp 325– (1989) · Zbl 0732.26009 · doi:10.1017/S0305004100078142
[4] DOI: 10.2307/2047947 · Zbl 0721.28004 · doi:10.2307/2047947
[5] Taylor, Math. Proc. Cambridge Phil. Soc 100 pp 383– (1986)
[6] DOI: 10.1512/iumj.1992.41.41031 · Zbl 0757.28011 · doi:10.1512/iumj.1992.41.41031
[7] Kono, Japan J. Appl. Math 3 pp 259– (1986)
[8] McMullen, Nagoya Math. J 96 pp 1– (1984) · Zbl 0539.28003 · doi:10.1017/S0027763000021085
[9] DOI: 10.1512/iumj.1981.30.30055 · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[10] DOI: 10.2307/2282952 · Zbl 0127.10602 · doi:10.2307/2282952
[11] Rogers, Mathematika 8 pp 1– (1961)
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