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

DOI:
https://doi.org/10.1090/S0002-9939-1990-1000169-8

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 are studied.

**[F]**K. J. Falconer,*The geometry of fractal sets*, Cambridge University Press, 1985. MR**867284 (88d:28001)****[K1]**N. Kôno,*On self-affine functions*, Japan J. Appl. Math.**3**(1986), 271-280. MR**899224 (88i:26014)****[K2]**-,*On self-affine functions*II, Japan J. Appl. Math.**5**(1988), 441-454. MR**965874 (90c:26024)****[M]**C. McMullen,*The Hausdorff dimension of general Sierpiński carpets*, Nagoya Math. J.**96**(1984), 1-9. MR**771063 (86h:11061)****[R]**C. A. Rogers,*Hausdorff measures*, Cambridge University Press, 1970. MR**0281862 (43:7576)****[S]**V. Strassen,*An invariant principle for the law of the iterated logarithm*, Z. Wahrscheinlichkeitstheorie**3**(1964), 211-226. MR**0175194 (30:5379)****[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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DOI:
https://doi.org/10.1090/S0002-9939-1990-1000169-8

Article copyright:
© Copyright 1990
American Mathematical Society