Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

In this talk I will present a result regarding the perturbation of rational maps satisfying the Collet--Eckmann condition, and also a certain recurrence condition for the set of critical points. If the Julia set is not the entire Riemann sphere, it turns out that these maps can be strongly approximated by hyperbolic maps. This results contrasts the situation when the Julia set is is the entire sphere since, in this case, it turns out that such maps are strongly approximated by Collet--Eckmann maps. This is a joint work together with Magnus Aspenberg and Weiwei Cui.