Non-recurrent dynamics in the exponential family

Abstract

This dissertation deals with the dynamics of non-recurrent parameters in the exponential family $\{e^z+c\}$. 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.