On the Dynamics of a Third Order Newton's Approximation Method

We show that the third order approximation function $M_f$, proposed by S. Amat, S. Busquier, S. Plaza, in $\mathit{\text{J.}}\mathit{\text{Math.}}\mathit{\text{Anal.}}\mathit{\text{Appl.}}$, 366(2010), 24–32, for functions $f$ twice continuously differentiable and such that both $f$ and its derivative do not have multiple roots, with at least four roots, and infinite limits of opposite signs at $\pm \mathrm{\infty }$, have periodic points of any prime period and that the set of points $a$ at which the approximation sequence $(M_f^n(a){)}_{n\in \mathbb{N}}$ does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.