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The Hausdorff dimension of the graphs of continuous self-affine functions


Author: M. Urbański
Journal: Proc. Amer. Math. Soc. 108 (1990), 921-930
MSC: Primary 26A30; Secondary 28A75
MathSciNet review: 1000169
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Abstract: The exact formula for the Hausdorff dimension of the graph of a continuous self-affine function is obtained. The Hausdorff dimension of some class of Borel probability measures is computed. The Hausdorff measures corresponding to the functions $ {\varphi _c}(t) = {t^{HD({\text{graph}}(f))}}\exp (c\sqrt {\log 1/t\log \log \log 1/t} $ are studied.


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

  • [F] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • [K1] Teturo Kamae, A characterization of self-affine functions, Japan J. Appl. Math. 3 (1986), no. 2, 271–280. MR 899224, 10.1007/BF03167102
  • [K2] Norio Kôno, On self-affine functions. II, Japan J. Appl. Math. 5 (1988), no. 3, 441–454. MR 965874, 10.1007/BF03167911
  • [M] Curt McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1–9. MR 771063
  • [R] C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
  • [S] V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 0175194
  • [U] M. Urbański, The probability distribution and Hausdorff dimension of self-affine functions, (to appear in Prob. Th. and Rel. Fields).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1000169-8
Article copyright: © Copyright 1990 American Mathematical Society