The iteration of cubic polynomials. I: The global topology of parameter space. (English) Zbl 0668.30008

Some decompositions of the space of complex polynomials according to their dynamical properties are described. It continues the earlier works of Douady and Hubbard concerning polynomials of degree 2. Let \({\mathcal P}_ d\) be the space of complex polynomials of degree \(d\geq 2\) of the form \[ P(z)=z^ d+b_ 2z^{d-2}+...+b_ d. \] For a polynomial \(P\in {\mathcal P}_ d\) denote by \(K_ P\) the filled in Julia set, \(h_ P(z)\) the Green function of \({\mathbb{C}}\setminus K_ P\) with a pole at \(\infty\). Let us consider the following function on the space \({\mathcal P}_ d:H(P)=\max \{h_ P(a_ i)| a_ i\) is a critical point of \(P\}\).
The paper consists of three chapters. It is proved in Chapter I that the sets \(\{\) \(P| H(P)\leq r\}\) are compact for all \(r\), and the level sets \({\mathcal S}_ r=\{P| H(P)=r\}\) are homeomorphic to the sphere \(S^{2d-3}\) for sufficiently large r. It follows that the connectedness locus (the Mandelbrot-like set) \({\mathcal C}_ d=\{P| K_ P\) is \(connected\}=\{P| H(P)=0\}\) is compact as well. The proofs are based upon geometric function theory (distortion theorems) applied to the Böttcher function \(\phi_ P\) in the neighbourhood of \(\infty.\)
In Chapter II a fine action of the group \(G=\left\{\begin{pmatrix} \tau & 0 \\ s & 1 \end{pmatrix} | \tau >0,s\in {\mathbb{R}}\right\}\) on \({\mathcal P}_ d\) is described. It is obtained by wringing the complex structure on \({\mathbb{C}}\setminus K_ P\) and considering the corresponding quasiconformal deformation of \(P\in {\mathcal P}_ d\). The level sets \({\mathcal S}_ r\) are transitively permitted by the action. For \(d=3\) th authors prove that the action is continuous in both \(P\in {\mathcal P}_ 3\) and \(g\in G\). It follows that H:\({\mathcal P}_ 3\setminus {\mathcal C}_ 3\to {\mathbb{R}}_+\) is a trivial fibration with fibres \({\mathcal S}_ r\) homeomorphic to the sphere \(S^ 3\). Consequently, the Mandelbrot-like set \({\mathcal C}_ 3\) is connected as well. The rest of the paper is devoted to the decomposition of spheres \({\mathcal S}_ r\) into the level sets of the function \(\psi(P)=\phi_ P(P(a))\) where a is the critical point of P fastest escaping to \(\infty\). It is proved that the level sets of \(\psi\) are homeomorphic to the trefoil clover leaf (the closed discs with one boundary point in common).
Reviewer: M.Lyubich


30C20 Conformal mappings of special domains
30C62 Quasiconformal mappings in the complex plane
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics


Zbl 0618.30026
Full Text: DOI


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