Holomorphic motions of hyperbolic nonautonomous Julia sets

Abstract

We generalize a number of the results of Mañé, Sad, and Sullivan concerning families of analytic functions depending on a parameter. The new results are in the context of nonautonomous iteration where instead of one polynomial or rational function, one considers iterates arising from an arbitrarily chosen sequence of polynomials. We use the famous λ-lemma to show that in a parameter neigbourhood of a sequence whose associated dynamics is hyperbolic, the Julia set moves smoothly and indeed holomorphically, much as in the original situation, even when one considers the question of holomorphic dependence on infinitely many independent complex parameters. Finally, we consider the parameter space associated with sequences of quadratic polynomials and show how our results imply that the structure is significantly different from that of quadratic polynomials in standard complex dynamics.