Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A Mandelbrot set whose boundary is piecewise smooth

Authors: M. F. Barnsley and D. P. Hardin
Journal: Trans. Amer. Math. Soc. 315 (1989), 641-659
MSC: Primary 58F08; Secondary 30D05
MathSciNet review: 1011232
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [1] B. Mandelbrot, Fractal aspects of the iteration of $ z \to \lambda z \cdot (1 - z)$, Ann. New York Acad. Sci. 357 (1980), 249-259; On the quadratic mapping $ z \to {z^2} - \mu $ for complex $ \mu $ and $ z$: the fractal structure of its $ M$ set and scaling, Physica 7D (1983), 224-239; On the dynamics of iterated maps VIII, Chaos and Statistical Methods (Y. Kuramoto, ed.), Springer, Berlin, 1984, pp. 32-42. MR 719054 (85d:58065)
  • [2] A. Douady and J. Hubbard, C. R. Acad. Sci. Paris 294 (1982), 123-126. MR 651802 (83m:58046)
  • [3] A. Douady, Systèmes dynamique holomorphes, Sem. Bourbaki 35 (599) (1982/1983).
  • [4] M. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), 25-52; P. J. Myrberg, Sur l'iteration des polynomes réels quadratiques, J. Math. Pures Appl. (9) 41 (1962), 339-351. MR 0501179 (58:18601)
  • [5] M. F. Barnsley and A. N. Harrington, Physica 15D (1985), 421-432. MR 793899 (86j:58088)
  • [6] J. Hutchinson, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [7] M F. Barnsley and S. G. Demko, Iterated function systems and global construction of fractals, Proc. Roy. Soc. London Ser. A 399 (1985), 243-275. MR 799111 (87c:58051)
  • [8] B. Mandelbrot, The fractal geometry of nature, Freeman, San Francisco, Calif., 1983. MR 665254 (84h:00021)
  • [9] H. Yoshida, Self-similar natural boundaries of nonintegrable dynamical systems in the complex $ t$-plane, Preprint, Dept. of Astronomy, Univ. of Tokyo. MR 771493
  • [10] 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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F08, 30D05

Retrieve articles in all journals with MSC: 58F08, 30D05

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society