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 \textit{J. Math. Anal. 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 $\pm\infty$, 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.