Exotic Baker and wandering domains for Ahlfors islands maps

TL;DR

This paper constructs examples of Ahlfors islands maps on Riemann surfaces of genus at most 1, demonstrating complex dynamical behaviors like Baker domains and wandering domains, with implications for the convergence rates of iterates.

Contribution

It provides new explicit constructions of Ahlfors islands maps exhibiting Baker domains, wandering domains, and specific boundary behaviors on genus 0 and 1 surfaces.

Findings

Existence of Baker domains with prescribed omega-limit sets

Entire functions with Baker domains where iterates converge arbitrarily slowly

Construction of Ahlfors islands maps with wandering domains and logarithmic singularities

Abstract

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.