It is known in many cases that the dynamics of a rational self-map of the complex plane is broadly governed by two numerical invariants, the first and second dynamical degrees of the map. The case when the degrees are equal, however, is fairly unexamined. Here I will discuss some recent work with Roland Roeder and Han Liu about precisely this situation. The aim is to give a picture of the dynamics of a plan rational map with equal degrees up to generic perturbation by linear maps. Speaker(s): Jeff Diller (University of Notre Dame)