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On Weierstrass-like functions and random recurrent sets

Published online by Cambridge University Press:  28 June 2011

Tim Bedford
Tim Bedford
Affiliation:
Department of Mathematics and Informatics, Delft University of Technology, P.O. Box 356, 2600 AJ Delft, The Netherlands
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Abstract

A construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 325 - 342
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Bedford, T.. Crinkly curves, Markov partitions and dimension. Ph.D. Thesis, University of Warwick, 1984.Google Scholar
[2] Billingsley, P.. Ergodic Theory of Information (Wiley, 1965).Google Scholar
[3] Bougerol, P. and Lacroix, J.. Products of Random Matrices with Application to Schrödinger Operators. Progr. Probab. Statist. (Birkhäuser, 1985).CrossRefGoogle Scholar
[4] Besicovitch, A. S. and Ursell, H. D.. Sets of fractional dimensions (V): on dimensional numbers of some continuous curves. J. London Math. Soc. 12 (1937), 18–25.CrossRefGoogle Scholar
[5] Dekking, F. M.. Recurrent sets. Adv. in Math. 44 (1982), 78–104.CrossRefGoogle Scholar
[6] Eggleston, H. G.. The fractional dimension of set defined by decimal properties. Quart. J. Math. Oxford Ser. (2) 20 (1949), 31–39.CrossRefGoogle Scholar
[7] Falconer, K. J.. Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986), 559–582.CrossRefGoogle Scholar
[8] Kono, N.. On self-affine functions II. Japan J. Appl. Math. 5 (1988), 441–454.CrossRefGoogle Scholar
[9] Marstrand, J. M.. The dimension of Cartesian product sets. Proc. Cambridge Philos. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[10] McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[11] Mauldin, R. D. and Williams, S. C.. Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295 (1986), 325–345.CrossRefGoogle Scholar
[12] Mauldin, R. D. and Williams, S. C.. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc. 298 (1986), 793–803.CrossRefGoogle Scholar