Critical recurrence in real quadratic and rational dynamics

In this thesis we study the dynamics of real quadratic functions on the interval, and rational functions on the Riemann sphere. The problems we are considering are concerned with the recurrent nature of the critical orbit(s). In Paper I we investigate the real quadratic family and prove a theorem regarding the rate of recurrence of the critical point to itself, extending a previous result by Avila and Moreira. In Paper II and Paper III we consider rational functions. Here we do not study the rate of recurrence, rather we assume that the critical points approach each other at a slow rate, and investigate some of the consequences. Assuming this slow recurrence condition, we prove in Paper II that certain Collet--Eckmann rational functions can in a strong sense be approximated by hyperbolic ones. In Paper III we observe that within the family of slowly recurrent rational maps, the well-known Collet--Eckmann conditions are all equivalent.