Rational maps lacking certain periodic orbits

This thesis is a study of the dynamics of rational maps without certain periodic points, focusing on the roles of critical points. We examine how the dynamics occurring within the parameterized family of exceptional rational maps are reected in the parameter space. We use Baker's results on the multiplicities of fixed points of a rational map to give a complete proof of the classification of exceptional rational maps. In degree 2 and 3 cases exceptional rational maps form parameterized families, all of whose members have parabolic fixed point(s). In the theory of the iteration of rational maps, the behavior of the critical points of a rational map determines the entire dynamics. We produce the parameter space pictures of the parameterized families of exceptional rational maps by tracking the orbits of critical points. Then We obtain the reduced regions using conformal conjugacy. We compare the parameterized family of quadratic exceptional maps and the family of quadratic polynomials focusing on the behavior of the critical points. We show that for parameter values on part of the boundary of the reduced region, in the dynamical plane there is a forward invariant circle and on it lie fixed point(s) as well as critical points