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.