Dynamics and limiting behavior of Julia sets of König's method for multiple roots

A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set Sd depending only on d such that, given any root α of f, there exists at least one point in Sd converging under iterations of Nf to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton's method. We show that for all order σ≥2, there exists a complex polynomial f such that the Julia set of König's method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König's method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric.