In this talk I will introduce an important class of non-hyperbolic rational maps on the Riemann sphere -- the so-called Collet--Eckmann maps -- and discuss how well these maps can be approximated by hyperbolic maps. In the rational setting, if the Julia set is not the entire sphere, and assuming a certain recurrence condition for the set of critical points, we show that these maps can be strongly approximated by hyperbolic maps. Indeed, in the parameter space, these maps are density points of hyperbolic maps (of the same degree). This results contrasts the situation when the Julia set is assumed to be the entire sphere since, in this case, it turns out that such maps are density points of Collet--Eckmann maps. I will also discuss a recent result for the unicritical family, where the same conclusion holds true, but without the assumption on recurrence. This talk is based on joint work together with Magnus Aspenberg and Weiwei Cui.
