Étude dynamique des polynômes complexes. (French) Zbl 0552.30018

Publ. Math. Orsay 84-02, 75 p. (1984).
These notes expound some of the results announced by the authors [C. R. Acad. Sci., Paris, Sér. I 294, 123-125 (1982; Zbl 0483.30014)] and sketched by the first author in Sémin. Bourbaki, 35e anneé, Vol. 1980/81, Exp. No.599, Astérisque 105/106, 39-63 (1983; Zbl 0532.30019). They deal with the iterates \(f^ n\) of \(f: z\to z^ 2+c,\) where c is a complex parameter, the filled in Julia set \(K(c)=\{z;\quad f^ n(z)\) is bounded\(\}\), the Mandelbrojt set (which is connected): \(M=\{c;\quad f^ n(0)\) is bounded\(\}\), and \(M_ 1=\{c\); f has an attractive cycle of some order\(\}\). Two outstanding conjectures state that I: M is locally connected, II: \(\overset\circ M=M_ 1\). In this and the second part the author proposes to show that I implies II. Much of the exposition is devoted to the Hubbard tree, a combinatorial scheme which can be constructed in K(c) in the cases when 0 is either a periodic point or the preimage of a periodic point of f.
Reviewer: I.N.Baker


30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37G99 Local and nonlocal bifurcation theory for dynamical systems