A Mandelbrot set whose boundary is piecewise smooth
HTML articles powered by AMS MathViewer
- by M. F. Barnsley and D. P. Hardin
- Trans. Amer. Math. Soc. 315 (1989), 641-659
- DOI: https://doi.org/10.1090/S0002-9947-1989-1011232-6
- PDF | Request permission
Abstract:
It is proved that the Mandelbrot set associated with the pair of maps ${w_{1,2}}:{\mathbf {C}} \to {\mathbf {C}}, {w_1}(z) = sz + 1, {w_2}(z) = {s^\ast }z - 1$, with parameter $s \in {\mathbf {C}}$, is connected and has piecewise smooth boundary.References
- Benoit B. Mandelbrot, On the quadratic mapping $z\rightarrow z^{2}-\mu$ for complex $\mu$ and $z$: the fractal structure of its ${\cal M}$ set, and scaling, Phys. D 7 (1983), no. 1-3, 224–239. Order in chaos (Los Alamos, N.M., 1982). MR 719054, DOI 10.1016/0167-2789(83)90128-8
- Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123–126 (French, with English summary). MR 651802 A. Douady, Systèmes dynamique holomorphes, Sem. Bourbaki 35 (599) (1982/1983).
- Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, DOI 10.1007/BF01020332
- M. F. Barnsley and A. N. Harrington, A Mandelbrot set for pairs of linear maps, Phys. D 15 (1985), no. 3, 421–432. MR 793899, DOI 10.1016/S0167-2789(85)80008-7
- 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
- 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
- Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR 665254
- H. Yoshida, Self-similar natural boundaries of nonintegrable dynamical systems in the complex $t$ plane, Chaos and statistical methods (Kyoto, 1983) Springer Ser. Synergetics, vol. 24, Springer, Berlin, 1984, pp. 42–45. MR 771493, DOI 10.1007/978-3-642-69559-9_{5} D. Bessis and N. Chafee, On the existence and non-existence of natural boundaries for non-intregrable dynamical systems, Chaotic Dynamics and Fractals (M. F. Barnsley and S. G. Demko, eds.), Academic Press, New York, 1985.
Bibliographic Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 315 (1989), 641-659
- MSC: Primary 58F08; Secondary 30D05
- DOI: https://doi.org/10.1090/S0002-9947-1989-1011232-6
- MathSciNet review: 1011232