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

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.