Conjecturally almost every map in the space Rd of rational maps of degree d at least 2 is either hyperbolic or satisfies the Collet-Eckmann condition (CE). The CE-condition is a type of non-uniform hyperbolicity and was the main topic in my thesis from 2004, where I proved that the set of CE-maps has positive measure in Rd, if d is at least 2. This also gives a new proof of an earlier famous result by M. Rees, that the set of maps admitting an absolutely continuous invariant measure has positive measure in Rd. A related class of maps are the so called Misiurewicz maps, which are non-hyperbolic maps without parabolic cycles and where every critical point in the Julia set does not accumulate on any other critical point. These maps are a special type of CE-maps, but they have measure zero. On the contrary, the Hausdorff dimension of these maps is full, i.e. equal to the dimension of the space Rd (showed together with J. Graczyk). Hence in a sense critically non-recurrent dynamics is rare.

One can also consider so called semi-hyperbolic maps, studied by Carleson, Jones and Yoccoz. For a semi-hyperbolic map every critical point is non-recurrent and there are no parabolic cycles (they are also assumed to be non-hyperbolic). Hence they form a larger class than the Misiurewicz maps. Recently Graczyk and myself proved that in degree 2 these maps also have measure zero. Conjecturally the same is true in any degree at least 2.