Magnus Aspenberg, Lund: Dimension and measure for critically non-recurrent rational maps

Given a (non-hyperbolic) rational map f on the Riemann sphere, if the set of critical points, Crit(f), is non-recurrent on the Julia set and f does not have any 
parabolic cycles, then we say that the map satisfies the Misiurewicz condition. These maps have Lebesgue measure zero and full Hausdorff dimension, 
meaning that the dimension is equal to the dimension of the parameter space. If we weaken 
this condition, and consider (non-hyperbolic) maps f for which every
critical point on the Julia set is not recurrent (and still f has no parabolic cycles) we say that f is semi-hyperbolic, 
following the notion by Carleson, Jones and Yoccoz. I will present results about these maps and also discuss some open questions.