Abstract
Let X be a compact Riemann surface of genus at most 1, i.e., the Riemann sphere or a torus, and let W ⊊ X be an arbitrary domain. We construct a variety of examples of holomorphic functions g: W → X that satisfy Epstein’s Ahlfors islands property and that have “pathological” dynamical behaviour. In particular, we show that the accumulation set of any curve tending to the boundary of W can be realized as the ω-limit set of a Baker domain of such a function. We furthermore construct Ahlfors islands maps
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with wandering domains having prescribed ω-limit sets
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with logarithmic singularities having prescribed asymptotic curves
and also produce examples where X is a compact hyperbolic surface. As a corollary of our method, we construct transcendental entire functions with Baker domains in which the iterates tend to infinity arbitrarily slowly.
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The first author is supported by EPSRC fellowship EP/E052851/1.
Both authors were supported by the European CODY network.
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Rempe, L., Rippon, P.J. Exotic baker and wandering domains for Ahlfors islands maps. JAMA 117, 297–319 (2012). https://doi.org/10.1007/s11854-012-0023-5
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DOI: https://doi.org/10.1007/s11854-012-0023-5