Non-autonomous Julia sets with escaping critical points
Abstract
We consider non-autonomous iteration which is a generalization of standard polynomial iteration where we deal with Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. In this paper, we look at examples where all the critical points escape to infinity. In the classical case, any example of this type must be hyperbolic and there can be only one Fatou component, namely the basin at infinity. This result remains true in the non-autonomous case if we also require that the dynamics on the Julia set be hyperbolic or semi-hyperbolic. However, in general it fails and we exhibit three counterexamples of sequences of quadratic polynomials all of whose critical points escape but which have bounded Fatou components. © 2011 Taylor and Francis Group, LLC.