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

Author: John McLaughlin
Journal: Proc. Amer. Math. Soc. 100 (1987), 183-186
MSC: Primary 54H20; Secondary 28A75, 58F12
MathSciNet review: 883425
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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 $ \S1$ the positivity is established for compact sets, and a lower bound is given for their Hausdorff measure. However, in $ \S2$ the finiteness is disproved. In fact, a quasi-self-similar set is constructed for which the Hausdorff measure is actually $ \sigma $-infinite.

References [Enhancements On Off] (What's this?)

  • [1] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • [2] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [3] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, 10.1512/iumj.1981.30.30055
  • [4] D. Sullivan, Seminar on conformal and hyperbolic geometry, Lecture Notes, Inst. Hautes Études Sci., Bures-sur-Yvette, 1982.

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Keywords: Self-similarity Hausdorff measure
Article copyright: © Copyright 1987 American Mathematical Society