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A note on Hausdorff measures of quasi-self-similar sets. (English) Zbl 0629.28006

D. Sullivan has demonstrated that quasi-self-similarity provides a useful point of view for the study of expanding dynamical systems. In “Seminar on conformal and hyperbolic geometry” [Lect. Notes, Inst. Hautes Etudes Sci., Bures-sur-Yvette (1982)] 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 § 1 the positivity is established for compact sets, and a lower bound is given for their Hausdorff measure. However, in § 2 the finiteness is disproved. In fact, a quasi-self-similar set is constructed for which the Hausdorff measure is actually \(\sigma\)-infinite.

MSC:

28A75 Length, area, volume, other geometric measure theory
54H20 Topological dynamics (MSC2010)
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References:

[1] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. · Zbl 0587.28004
[2] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[3] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 – 747. · Zbl 0598.28011
[4] D. Sullivan, Seminar on conformal and hyperbolic geometry, Lecture Notes, Inst. Hautes √Čtudes Sci., Bures-sur-Yvette, 1982.
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