The secant map as a plane dynamical system

We present some results on the rational dynamical system induced by the secant map on the real plane. The secant map is a root finding algorithm which does not use the evaluation of the derivative of the corresponding map. Here we apply the secant map to real polynomials and we are interested only on its real roots (if any).

On the one hand we prove that there exist some points, called focal points, belonging to the boundary of the basins of attraction of all  fixed points. On the other hand we prove the existence of polynomials for which there are open sets of the plane not convergeging to any of its roots. Finally we will discuss some further dynamical considerations by somehow adding infinity on the field.