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Transactions of the American Mathematical Society

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Random recursive constructions: asymptotic geometric and topological properties


Authors: R. Daniel Mauldin and S. C. Williams
Journal: Trans. Amer. Math. Soc. 295 (1986), 325-346
MSC: Primary 60D05; Secondary 28A75, 54H20
DOI: https://doi.org/10.1090/S0002-9947-1986-0831202-5
MathSciNet review: 831202
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Abstract: We study some notions of "random recursive constructions" in Euclidean $ m$-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiński curve or Menger curve. We demonstrate that associated with each such construction is a "universal" number $ \alpha $ such that almost surely the random object has Hausdorff dimension $ \alpha $. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula.


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

  • [1] K. B. Athreya and P. E. Ney, Branching processes, Springer-Verlag, Berlin, 1972. MR 0373040 (51:9242)
  • [2] R. D. Anderson, A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313-324. MR 0096180 (20:2675)
  • [3] -, One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1-16. MR 0096181 (20:2676)
  • [4] J. W. Cannon, Book review, Amer. Math. Monthly 91 (1984), 594-598. MR 1540536
  • [5] L. E. Dubins and D. A. Freedman, Random distribution functions, Proc. Fifth Berkeley Sympos. on Math. Statistics and Probability (L. M. LeCam and J. Neyman, eds.), Univ. of California Press, Berkeley, Calif., 1967, pp. 183-214. MR 0214109 (35:4960)
  • [6] P. Doubilet, On the foundations of combinatorial theory. VIII: Symmetric functions through the theory of distribution and occupancy, Stud. Appl. Math. 51 (1972), 377-396. MR 0429577 (55:2589)
  • [7] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Math., vol. 85, Cambridge Univ. Press, Cambridge, 1985. MR 867284 (88d:28001)
  • [8] S. Graf, R. D. Mauldin and S. C. Williams, Random homeomorphisms, Adv. in Math. (to appear). MR 848153 (87k:60010)
  • [9] W. Hurewicz and H. Wallman, Dimension theory, Princeton Math. Series, vol. 4, Princeton Univ. Press, Princeton, N.J., 1941. MR 0006493 (3:312b)
  • [10] V. Jarńik, Über die stetigen Abbildungen der Strecke, Monatsh. Math. Phys. 41 (1934), 408-428. MR 1550323
  • [11] B. B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, Calif., 1982. MR 665254 (84h:00021)
  • [12] -, Fractals: form, chance and dimension, Freeman, San Francisco, Calif., 1977. MR 0471493 (57:11224)
  • [13] S. Mazurkiewicz, Über die stetigen Abbildungen der Strecke, Fund. Math. 25 (1935), 253-260.
  • [14] K. Menger, Kurventheorie, Chelsea, New York, 1967. MR 0221475 (36:4527)
  • [15] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15-23. MR 0014397 (7:278f)
  • [16] C. A. Rogers and S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure, Mathematika 8 (1961), 1-31. MR 0130336 (24:A200)
  • [17] W. Sierpiński, Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donée, C. R. Acad. Sci. Paris 162 (1916), 626-631.
  • [18] G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math. 45 (1958), 320-324. MR 0099638 (20:6077)
  • [19] U. Zähle, Random fractals generated by random cutouts, Math. Nachr. 116 (1984), 27-52. MR 762590 (87b:60018)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0831202-5
Keywords: Martingale, Hausdorff dimension, Cantor set, Sierpiński curve, Menger curve
Article copyright: © Copyright 1986 American Mathematical Society

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