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Iteration of meromorphic functions
This paper attempts to describe some of the results obtained in the iteration
theory of transcendental meromorphic functions, not excluding the case of
entire functions. The reader is not expected to be familiar with the iteration
theory of rational functions. On the other hand, some aspects where the
transcendental case is analogous to the rational case are treated rather
briefly here. For example, we introduce the different types of components of
the Fatou set that occur in the iteration of rational functions but omit a
detailed description of these types. Instead, we concentrate on the types of
components that are special to transcendental functions (Baker domains and
wandering domains).Comment: 38 pages. Abstract added in migration. See
http://analysis.math.uni-kiel.de/bergweiler/ for recent comments and
correction
Exotic Baker and wandering domains for Ahlfors islands maps
Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere
or a torus. We construct a variety of examples of analytic functions g:W->X,
where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands
condition". In particular, we show that the accumulation set of any curve
tending to the boundary of W can be realized as the omega-limit set of a Baker
domain of such a function. As a corollary of our construction, we show that
there are entire functions with Baker domains in which the iterates converge to
infinity arbitrarily slowly. We also construct Ahlfors islands maps with
wandering domains and logarithmic singularities, as well as examples where X is
a compact hyperbolic surface.Comment: 18 page
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