Exotic Baker and wandering domains of Ahlfors islands functions

With Phil Rippon


Bibliographic data

J. Anal. Math. 117 (2012), 297–319.
DOI: 10.1007/s11854-012-0023-5
Preprint version at arxiv:1008.1724.

Short Summary

This paper constructs examples of functions satisfying Adam Epstein's "Ahlfors islands property" (see ""Hyperbolic dimension and radial Julia sets") which have various interesting dynamical or function-theoretic properties, using approximation theory.

If X is the Riemann sphere or a torus, and W is a proper subdomain of X, then we show that there are Ahlfors islands maps g:W->X. Moreover, these can be chosen such that:

a) g has a Baker domain whose set of limit functions coincides with any prescribed compact connected subset of the boundary of W that can be written as the accumulation set of a curve in W; moreover the iterates in the Baker domain can be chosen to tend to infinity arbitrarily slowly;
b) g has a wandering domain whose set of limit functions coincides with any prescribed compact subset of the boundary of W;
c) given a prescribed C^1 curve gamma tending to the boundary, g can be constructed with a logarithmic asymptotic value that has gamma as an asymptotic curve.

Note that c) is a nondynamical statement, and in fact W can be replaced by any proper subdomain of any compact Riemann surface Y. In particular, it follows that there are no restrictions on the possible domains of Ahlfors islands functions when the target Riemann surface has genus at most 1.

While most of the results are well-known when X is the Riemann sphere and W is the complex plane or the punctured plane, the statement about escape speeds in a) also yields an apparently new result for entire functions: The rate of escape in a Baker domain of a transcendental entire function can be arbitrarily slow.

The construction uses Arakelian's approximation theorem and its generalization to analytic functions on Riemann surfaces due to Scheinberg. In order to be able to construct Baker domains, we need to establish approximation results for functions taking values in a simply-connected domain that are a priori stronger than those provided by the Arakelian-Scheinberg theorem.

We also construct examples as in a), b) and c) above when X is a compact hyperbolic Riemann surface. However, in this case the constructed function will be an Ahlfors islands map g:W'->X with the stated properties, where W' is a proper subdomain of W.