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