Newton's method applied to rational functions: Fixed points and Julia sets

For a rational function R, let NR(z)=z-R(z)/R'(z). Any such NR is referred to as a Newton map. We determine all the rational functions R for which NR has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier $2$, or the multiplier of the non-exceptional attracting fixed point is at most 4/5, then its Julia set is shown to be connected. If a polynomial p has exactly two roots, is unicritical but not a monomial, or p(z)=z(zⁿ+a) for some a ∈ C and n ≥ 1, then we have proved that the Julia set of N1/p is totally disconnected. For the McMullen map fλ(z)=zᵐ - λ/zⁿ, λ ∈ C \0\ and m,n ≥ 1, we have proved that the Julia set of Nf_λ is connected and is invariant under rotations about the origin of order $m+n$. All the connected Julia sets mentioned above are found to be locally connected.