Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions (Q1088025)

The Schröder and König iteration schemes to find the zeros of a (polynomial) function g(z) represent generalizations of Newton's method. In both schemes, iteration functions \(f_ m(z)\) are constructed so that sequences \(z_{n+1}=f_ m(z_ n)\) converge locally to a root \(z^ *\) of g(z) as \(O(| z_ n-z^ *| ^ m)\). It is well known that attractive cycles, other than the zeros \(z^ *\), may exist for Newton's method \((m=2)\). As m increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functions \(K_ m(z)\) appear to minimize such perturbations. In the case of two roots, e.g. \(g(z)=z^ 2- 1\), Cayley's classical result for the basins of attraction of Newton's method is extended for all \(K_ m(z)\). The existence of chaotic \(\{z_ n\}\) sequences is also demonstrated for these iteration methods.