On Hausdorff dimension of recurrent net fractals
HTML articles powered by AMS MathViewer
- by Sergio Stella
- Proc. Amer. Math. Soc. 116 (1992), 389-400
- DOI: https://doi.org/10.1090/S0002-9939-1992-1094507-X
- PDF | Request permission
Corrigendum: Proc. Amer. Math. Soc. 121 (1994), 1309-1311.
Abstract:
We estimate the Hausdorff dimension of recurrent net fractals showing that it coincides with the box-counting dimension. This is done for geometric constructions in a complete metric space, generalizing well-known theorems about self-similar sets. In particular, it follows that what is really essential in the dimension estimates of self-similar sets are their metric features and not the dynamical ones.References
- Michael F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1986), no. 4, 303–329. MR 892158, DOI 10.1007/BF01893434
- M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), no. 1817, 243–275. MR 799111
- Michael F. Barnsley, John H. Elton, and Douglas P. Hardin, Recurrent iterated function systems, Constr. Approx. 5 (1989), no. 1, 3–31. Fractal approximation. MR 982722, DOI 10.1007/BF01889596 T. Bedford, Crinkly curves, Markov partitions and dimensions, Ph.D. Thesis, Warwick University, Warwick, 1984.
- Tim Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), no. 1, 89–100. MR 829390, DOI 10.1112/jlms/s2-33.1.89
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
- K. J. Falconer, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 559–582. MR 857731, DOI 10.1017/S0305004100066299
- K. J. Falconer, Dimensions and measures of quasi self-similar sets, Proc. Amer. Math. Soc. 106 (1989), no. 2, 543–554. MR 969315, DOI 10.1090/S0002-9939-1989-0969315-8
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- D. G. Larman, A new theory of dimension, Proc. London Math. Soc. (3) 17 (1967), 178–192. MR 203691, DOI 10.1112/plms/s3-17.1.178 B. B. Mandelbrot, The fractal geometry of nature, Freemann, New York, 1983.
- R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
- P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. MR 14397, DOI 10.1017/s0305004100022684
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. MR 2209438
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 389-400
- MSC: Primary 58F12; Secondary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-1992-1094507-X
- MathSciNet review: 1094507