Schröder iteration functions associated with a one-parameter family of biquadratic polynomials

Abstract

Schröder iteration functions, a generalization of the Newton–Raphson method to determine roots of equations, are generally rational functions which possess some critical points, free to converge to attracting cycles. These free critical points, however, satisfy some higher-degree polynomial equations which we solve analytically. Then, with the help of microcomputer plots, we examine the Julia sets of the Schröder functions and the orbits of all their free critical points associated with a particular one-parameter family of quartic polynomials, by walking in their dynamic and parameter spaces. This examination takes place in the complex plane as well as on the Riemann sphere.