Boundedness and simple connectivity of the basins of attraction for some numerical methods

In this paper we study the dynamics of Halley's and Traub's root-finding algorithms applied to a symmetric family of polynomials of degree $d+1\geq 3$. We discuss the (un)boundedness and simple connectivity of the immediate basins of attraction of the fixed points associated to the roots of the polynomials. In particular, we show the existence of polynomials for which the immediate basin of attraction of a root is bounded under Halley's method