Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study

Schröder iteration functions ${S_m}(z)$, a generalization of Newton's method (for which $m = 2$), are constructed so that the sequence ${z_{n + 1}} = {S_m}({z_n})$ converges locally to a root ${z^\ast}$ of $g(z) = 0$ as $O(\vert{z_n} - {z^\ast}{\vert^m})$. For $g(z)$ a polynomial, this involves the iteration of rational functions over the complex Riemann sphere, which is described by the classical theory of Julia and Fatou and subsequent developments. The Julia sets for the ${S_m}(z)$, as applied to the simple cases ${g_n}(z) = {z^n} - 1$, are examined for increasing m with the help of microcomputer plots. The possible types of behavior of ${z_n}$ iteration sequences are catalogued by examining the orbits of free critical points of the ${S_m}(z)$, as applied to a one-parameter family of cubic polynomials.