Convergence acceleration as a dynamical system (Q1339333)

The dynamical system \(z \to f(z)\) is replaced by \(z \to F_ n(z)\) where \(F_ n\) is given by a general Shanks transformation, i.e. a punctual diagonal Padé approximation leading, in fact, to the quotients of two Hankel or Toeplitz determinants. The particular case of the usual Aitken transformation and its improved form by Steffensen is studied. The author shows that \(F_ n\) retains all finite fixed points of \(f\). Moreover, for a polynomial \(f\) the most poles and zeros of \(F_ n\) accumulate on and inside the Julia set of \(f\) as \(n \to \infty\).