Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell
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.
KeywordsBifurcation Point Stability Domain Extensive Result Circular Boundary Figure Plot
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