A note on Hausdorff measures of quasi-self-similar sets

McLaughlin, John

doi:10.1090/S0002-9939-1987-0883425-3

A note on Hausdorff measures of quasi-self-similar sets
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by John McLaughlin

Proc. Amer. Math. Soc. 100 (1987), 183-186

DOI: https://doi.org/10.1090/S0002-9939-1987-0883425-3

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Abstract:

Sullivan has demonstrated that quasi-self-similarity provides a useful point of view for the study of expanding dynamical systems. In [4, p. 57] he posed the question: Is the Hausdorff measure of a quasi-self-similar set positive and finite in its Hausdorff dimension? This paper answers both parts of this question. In $\S 1$ the positivity is established for compact sets, and a lower bound is given for their Hausdorff measure. However, in $\S 2$ the finiteness is disproved. In fact, a quasi-self-similar set is constructed for which the Hausdorff measure is actually $\sigma$-infinite.

References

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John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055

Seminar on conformal and hyperbolic geometry