FRACTAL ASPECTS OF THE ITERATION OF z →Λz(1‐ z) FOR COMPLEX Λ AND z

@article{Mandelbrot1980FRACTALAO,
  title={FRACTAL ASPECTS OF THE ITERATION OF z →$\Lambda$z(1‐ z) FOR COMPLEX $\Lambda$ AND z},
  author={Benoit B. Mandelbrot},
  journal={Annals of the New York Academy of Sciences},
  year={1980},
  volume={357}
}
  • B. Mandelbrot
  • Published 1980
  • Mathematics
  • Annals of the New York Academy of Sciences
Chapter foreword concerning the illustrations, especially the “missing specks” of Figure 1 (2003). As described in Chapter C1, this paper boasts many “firsts” and was instrumental in reviving the theory of iteration. The many new observations it contains concern the set in the μ-plane for which A. Douady and J.H. Hubbard soon proposed the term “Mandelbrot set.” Each observation was stated as a mathematical conjecture or became the source of one. Thus, the figures in this paper played a… Expand
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References

SHOWING 1-10 OF 10 REFERENCES
Fractals: Form, Chance, and Dimension.
Some people may be laughing when looking at you reading in your spare time. Some may be admired of you. And some may want be like you who have reading hobby. What about your own feel? Have you feltExpand
Sur les équations fonctionnelles
© Bulletin de la S. M. F., 1920, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accordExpand
Self inverse fractals and Kleinian groups
  • Mathematical Intelligencer
  • 1980
Sur les Cquations fonctionnelles
  • Bull. SOC. Math. France
  • 1919
1919 - 1920 . Sur les Cquations fonctionnelles
  • Bull . SOC . Math . France
  • 1918
Mandelbrot: Fractal Aspects of Iteration of z -hz( 1 -z ) 259
    Sur lec solutions uniformes de certaines equations fonctionnelles
    • GUREL. 0. & 0. E. ROSSLER, I5ds. 1979. Bifurcation theory and applications in scientific FATOC, P. 1906