A note on Hausdorff measures of quasi-self-similar sets
HTML articles powered by AMS MathViewer
- by John McLaughlin PDF
- Proc. Amer. Math. Soc. 100 (1987), 183-186 Request permission
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
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- 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 D. Sullivan, Seminar on conformal and hyperbolic geometry, Lecture Notes, Inst. Hautes Études Sci., Bures-sur-Yvette, 1982.
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 183-186
- MSC: Primary 54H20; Secondary 28A75, 58F12
- DOI: https://doi.org/10.1090/S0002-9939-1987-0883425-3
- MathSciNet review: 883425