Extraneous fixed points of Euler iteration and corresponding Sullivan's basin (Q1609638)

The iteration function for the Newton method in the scalar case, that is, \[ Nf(z)=z-\frac{f(z)}{f'(z)}, \] has the property that any fixed point of \(Nf\) is a root of the equation (1) \(f(z)=0\), provided that \(f'\neq 0.\) If we consider Euler's iterative method, whose iteration function is given by \[ Ef(z)=z-\frac{f(z)}{f'(z)}-\frac{f''(z)f(z)^{2}}{ 2f'(z)^{3}}, \] then there exist fixed points (called extraneous) which are not roots of the equation (1). The paper constructs polynomials \(f(z)\) and initial values that produce extraneous fixed points by Euler's iterative method.