Abstract
A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper, we prove that the random self-conformal set is regular almost surely. Also we determine the dimensions for a class of random self-conformal sets.
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References
Arbeiter, M., Random Recursive Construction of Self-Similar Fractal Measure, The noncompact case. Probab. Th. Rel. Fields, 88(1990), 497–520.
Arbeiter, M. and Patzschke, N., Random Self-Similar Multifractals, Math. Nachr. 181(1996), 5–42.
Bedford, T., Hausdorff Dimension and Box Dimension in Self-Similar Sets, Proc. Conf. Topology and Measure V., Binz., GDR(1987), 17–23.
Berlinkov, A. and Mauldin, R. D., Packing Measure and Dimension of Random Fractals, Preprint.
Cawley, R. and Mauldin, R. D., Multifractal Decomposition of Moran Fractsls, Adv. in Math. 92(1992), 196–232.
Falconer, K. J., Random Fractals, Math. Proc. Cambridge Phli. Soc. 100(1986), 559–582.
Falconer, K. J., Dimensions and Measures of Quasi Self-Similar Sets, Proc. Amer. Math. Soc. 106(1989), 543–554.
Falconer, K. J., The Multifractal Spectrum of Statistically Self-Similar Measures, J. Theor. Prob. 7(1994), 681–702.
Falconer, K. J., Techniques in Fractal Geometry, John Wiley and Sons, Ltd, Chichester, 1997.
Graf, S., Statistically Self-Similar Fractals, Probab. th. Rel. Fields, 74(1987), 358–392.
Graf, S., Mauldin, R. D. and Williams, S. C., The Exact Hausdorff Dimension in Random Recursive Constructions, Mem. Amer. Math. Soc., 71(1988), no. 381.
Hutschinson, J. E., Fractal and Self-Similarity, Indiana Univ. Math. J. 30(1981), 713–747.
Hutschinson, J. E. and Ruschendorff, L., Random Fractal Measures via the Contraction Method, Indiana Univ. Math. J., 47(1998), 471–487.
Liang, J. R. and Ren, F. Y., Hausdorff Dimensions of Random net Fractals, Stoch. Process and their Appl., 74(1998), 235–250.
Liu, Y. Y. and Wu, J., A Dimensional Result for Random Self-Similar Sets, Preprint.
Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability. Cambridge University Press, 1995.
Mauldin, R. D. and Williams, S. C., Random Recursive Constuctions: Asymptotic Geometric and Topological Properties, Tran. Amer. Math. Soc. 295(1986), 325–346.
Patzschke, N., Self-Cconformal Multifractal Measure, Adv. Appl. Math. 19(1997), 486–513.
Patzschke, N., The Strong Open Set Ccondition in the Random Case, Proc. Amer. Math. Soc. 125(1997), 2119–2125.
Patzschke, N., Self-Conformal Multifractal Random Measure, Preprint.
Patzschke, N. and Zähle, U., Self-Similar Random Measures IV.—The Rrecursive Construction Model of Falconer, Graf, and Mauldin and Williams, Math. Nachr. 149(1990) 285–302.
Patzschke, N. and Zähle, M., Self-Similar Random Measures are Locally Scale Invariant, Probab. Th. Rel. Fields, 97(1993), 559–574.
Takens, F., Limit Capacity and Hausdorff Dimension of Dynamically Defined Cants, Lect. Notes in Math. 1331, Springer, 1988, 196–212.
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This paper was presented in the Fractal Satellite Conference of ICM 2002 in Nanjing.
The research was supported by the Special Funds for Major State Basic Research Projects in China.
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Yanyan, L., Jun, W. Dimensions for random self—Conformal sets. Anal. Theory Appl. 19, 342–354 (2003). https://doi.org/10.1007/BF02835533
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DOI: https://doi.org/10.1007/BF02835533