The construction of rational iterating functions

Abstract:

Suppose integers $n \geqslant 1$ and $\sigma \geqslant 2$ are given, together with n distinct points ${z_2}, \ldots ,{z_n}$, in the complex plane. Define ${\Phi _M} = {\Phi _M}(\sigma ;{z_1}, \ldots ,{z_n})$ to be the class of rational functions ${\phi _{p,q}}(z) = {g_p}(z)/{h_q}(z)$ (where g and h are polynomials of degree $p \geqslant 1$ and $q \geqslant 1$, respectively) such that $(1)\;p + q + 1 = M,(2)\;\phi$ when iterated converges with order $\sigma$ at each ${z_i}, i = 1, \ldots ,n$. Then if $M < \sigma n,{\Phi _M}$ is null; if $M = \sigma n$ ${\Phi _M}$ contains exactly $\sigma n$ elements. For every $M \geqslant \sigma n$, we show how to construct all the elements of ${\Phi _M}$ by expressing, for each choice of p and q which satisfies $p + q + 1 = M$ , the coefficients of ${g_p}$ and ${h_q}$ in terms of $M - \sigma n$ arbitrarily chosen values. In fact, these coefficients are expressed in terms of generalized Newton sums $S_n^{j,k} = S_n^{j,k}({z_1}, \ldots ,{z_n})$, $1 \leqslant j \leqslant n,k \geqslant n$, which we show may be calculated by recursion from the normal Newton sums $S_n^{j,n}$. Hence, given a polynomial ${f_n}(z)$ with n distinct (unknown) zeros ${z_1}, \ldots ,{z_n}$, we may construct all ${\phi _{p,q}}(z)$ which converge to the ${z_i}$ with order $\sigma$ in the case $\sigma = 2$, the choice $p = n$, $q = n - 1$, yields the Newton-Raphson iteration ${\phi _{n,n - 1}} \in {\Phi _{2n}}$; the Schröder and König iterations are shown to be elements of ${\Phi _{2(2\sigma - 3)(n - 1) + 2}}$ and ${\Phi _{2(\sigma - 1)(n - 1) + 2}}$, respectively. We show by example that there exist cases in which ${\phi _{n,n - 1}}$ has an undesirable property (attractive cycles) not shared by other iterating functions in the same class ${\Phi _{2n}}$.