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Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell

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Abstract

Foreword to this chapter and the appended figure (2003). The n 2 conjecture advanced in this chapter’s Section 2 was first proven in Guckenheimer & McGehee 1984. The two authors and I were participating in a special year on iteration that Lennart Carleson and Peter W. Jones convened during 1983–1984 at the Mittag-Leffler Institute in Djursholm (Sweden). During a seminar that I was giving, two auditors suddenly stopped listening and started writing furiously. After my talk ended, they rushed up with proofs that turned out to be identical and led to a joint report. They explained the n 2 phenomenon in terms of the normal forms of resonant bifurcations with multiplier exp(2πi/n). More extensive results establish that these stability domains have a limiting shape following rescaling. They are corollaries of the theory of analytic normal forms for parabolic points. See, for example, Shishikura 2000.

Keywords

  • Bifurcation Point
  • Stability Domain
  • Extensive Result
  • Circular Boundary
  • Figure Plot

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© 2004 Benoit B. Mandelbrot

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Mandelbrot, B.B. (2004). Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell. In: Fractals and Chaos. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4017-2_6

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  • DOI: https://doi.org/10.1007/978-1-4757-4017-2_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-1897-0

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