On the dynamics of a third order Newton's approximation method

Abstract

We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function M-f, by proving that 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 +/-infinity, M-f has periodic points of any prime period and that the set of points a at which the approximation sequence (M-f(n)(a))(n is an element of N) does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.