Slowly recurrent Collet–Eckmann maps on the Riemann sphere

Abstract

AbstractIn this paper we study perturbations of rational Collet–Eckmann maps for which the Julia set is the whole sphere, and for which the critical set is allowed to be slowly recurrent. Generically, if each critical point is simple, we show that each such Collet–Eckmann map is a Lebesgue point of Collet–Eckmann maps in the space of rational maps of the same degree $$d \ge 2$$ d ≥ 2 . The same result holds in each subspace, where we fix the multiplicities of the critical points.