| The theory of iterated transcendental functions has been extensively studied in the past two decades. We are interested in some "slices" of parameter spaces of certain classes of meromorphic functions with two asymptotic values, Ta,lambda, Slambda. We study the properties of the dynamic plane of functions in the families. We also study parametric representation of the families. We study the relationships between Slambda and the tangent family, between Slambda and the exponential family, between Ta,lambda and the tangent family and between Ta,lambda and the exponential family.;The functions Ta,lambda have two asymptotic values, one is -lambda and the other one is alambda. Under conjugation, the family can be written as {Ta,lambda (z) = alambda expz-exp -zexp z+aexp-z , a ∈ R , lambda ∈ C {0}}. We can see that as a approaches infinity, the asymptotic value alambda escapes to infinity, and that each function Ta,lambda(z), on any compact subset, will uniformly converge to the exponential function lambda exp(2z) - lambda. We will show that there is dynamic convergence as a → infinity, and we will study the relationship of the hyperbolic components of the two families, Ta,lambda and lambda exp(2z) - lambda.;In the family Slambda each function Slambda has two asymptotic values, 0 and lambda, and 0 is also a pole. We will show that each component of the Fatou set of Slambda is simply connected, and that there is at most one completely invariant domain of the Fatou set. We will also prove that these results can be generalized to functions with finitely many singular values and certain restrictions. |
