  In this report we construct a family of holomorphic functions
$\beta_{\lambda,\mu} (s)$ which behave asymptotically like iterated
exponentials as $|s| \to \infty$ in the right half plane. Each
$\beta_{\lambda,\mu}$ satisfies a convenient functional relationship with
nested exponentials; and has a series expansion that converges in a half-plane.
They provide a nearness to the dynamics of the map $e^{\mu z} :
\mathbb{C}\to\mathbb{C}$ and behave asymptotically as a fractional iteration
would behave.
  These objects are used to describe the various orbits of the exponential
function. We describe where Abel equations are feasibly constructed from
$\beta$. Where there exists wildly holomorphic functions with period $2 \pi i /
\lambda$ that are holomorphic Abel functions of the form $t(s+1) = e^{\mu
t(s)}$.

Complex Variables;Dynamical Systems