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.


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


[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.