99-4   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Iteration of inner functions and boundaries of components of the Fatou set

Let $D$ be an unbounded invariant component of the Fatou set of a transcendental entire function $f$. Let $\phi : \dz \to D$ be a Riemann map. Then the set $\Theta := \{ \theta \in \partial \dz \pl ; \lms \phi (r\theta ) = \infty \}$ is closely related to the Julia set of the corresponding inner function $g:= \phi^{-1} \circ f \circ \phi$. In the first part of the paper we further develop the theory of Julia sets of inner functions and the dynamical behaviour on their Fatou sets. In the second part we apply these results to iteration of entire functions by using the above relation and obtain some new results about the boundaries of components of the Fatou set of an entire function.

Mathematics Subject Classification (1991): 30D05, 58F08

Keywords: inner function, Julia set, Fatou set, boundary, radial limit, Baker domain.