Non-recurrent dynamics in the exponential family

This dissertation deals with the dynamics of non-recurrent parameters in the exponential family {ez + c}. One of the main open problems in one-dimensional complex dynamics is whether hyperbolic parameters are dense; this conjecture can be restated by saying that all fibers, i.e. classes of parameters with the same ray portrait, are single points unless they contain a hyperbolic parameter. The main goal of this dissertation was to prove some statements in this direction, usually referred to as rigidity statements.;We prove that fibers are single points for post-singularly finite (Misiurewicz) parameters and for combinatorially non-recurrent parameters with bounded post-singular set. We also prove some slightly different rigidity statement for combinatorially non-recurrent parameters with unbounded postsingular set.;We also add some understanding to the correspondence between combinatorics of polynomials and combinatorics of exponentials and we prove hyperbolicity of the postsingular set for non-recurrent parameters, generalizing a previous statement concerning only non-recurrent parameters with bounded post-singular set.;We finally contribute to another open problem in transcendental dynamics, i.e. understanding whether repelling periodic orbits are landing points of dynamic rays, giving a positive answer to this question in the case on non-recurrent parameters with bounded post-singular set.;The strategy used also gives a new, more elementary proof of the corresponding statement for polynomials, dating back to work of Douady.