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 [
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.
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The map
The dynamic plane for
The parameter plane for