Slices of parameter spaces of generalized Nevanlinna functions

Abstract

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps.Here, we extend these ideas to transcendental functions.In [16], it was shown that for the tangent family, $ \{ \lambda \tan z \} $, the hyperbolic components meet at a parameter $ \lambda^* $ such that $ f_{ \lambda^*}^n( \lambda^*i) = \infty $ for some $ n $. The behavior there reflects the dynamic behavior of $ \lambda^* \tan z $ at infinity. In Part 1. we show that this duality extends to a more general class of transcendental meromorphic functions $ \{f_{\lambda}\} $ for which infinity is not an asymptotic value. In particular, we show that in 'dynamically natural' one-dimensional slices of parameter space, there are 'hyperbolic-like' components $ \Omega $ with a unique distinguished boundary point such that for $ \lambda \in \Omega $, the dynamics of $ f_\lambda $ reflect the behavior of $ f_\lambda $ at infinity. Our main result is that every parameter point $ \lambda $ in such a slice for which the iterate of the asymptotic value of $ f_\lambda $ is a pole is such a distinguished boundary point.In the second part of the paper, we apply this result to the families $ \lambda \tan^p z^q $, $ p, q \in \mathbb Z^+ $, to prove that all hyperbolic components of period greater than $ 1 $ are bounded.