Branched coverings and cubic Newton maps. (English) Zbl 0903.58029

Tan, Lei

Fundam. Math. 154, No. 3, 207-260 (1997).

Let \(P\) be a cubic polynomial with simple zeros \(p_1\), \(p_2\) and \(p_3\); and \(N_P(z)= z-P(z)/P'(z)\). We denote a critical point of \(N_P\) other than \(p_1\), \(p_2\) and \(p_3\) by \(x_0\). A map \(N\) is called a cubic Newton map of \(P\) if \(N\) is a rational map conformally conjugate to \(N_P\). Furthermore, a cubic Newton map is called postcritically finite if \(x_0\) has a finite orbit.
The main theorem in this paper is the following. Let \(f(z)= z^3+ 3z/2\). There exists a subset \(A\) of cubic polynomials and a subset \(Y\) of the filled set of \(f\) such that there is a surjective mapping \({\mathcal M}\) from \(A\cup Y\) onto the set of postcritically finite cubic Newton maps such that for any \(g\in A\) (resp. \(y\in Y)\), the Newton map \({\mathcal M} (g)\) (resp. \({\mathcal M} (y))\) is equivalent to the mating \(f\coprod g\) (resp. the capture \(F_y)\).
Moreover, 1. any map \(N_P\) with \(x_0\) either periodic (but not fixed) or a preimage of \(p_1\) is equivalent to a unique mating; 2. any map \(N_P\) with \(x_0\) a preimage of \(p_2\) or \(p_3\) is equivalent to a capture; 3. any other postcritically finite \(N_P\) is equivalent to a capture and a mating with two choices of \(g\in A\) in the case that \(x_0\) is a preimage of \(\infty\), or a unique \(g\in A\) otherwise.

Reviewer: Z.Ye (DeKalb)

MSC:

37F99

Dynamical systems over complex numbers

Keywords:

cubic Newton map; branched covering; critically finite; matings and captures