Hyperbolic entire functions with bounded Fatou components

(joint work with Walter Bergweiler and Núria Fagella)


Bibliographical information

Preprint arxiv:1404.0925, 2014.

Abstract.

We show that an invariant Fatou component of a hyperbolic transcendental entire function is a bounded Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.

Short summary

We prove that an invariant Fatou component of a hyperbolic transcendental entire function is bounded if and only if it contains no asymptotic curves and only finitely many critical points. If this is the case, then the component is in fact a quasidisc.

Some applications are as follows:
  1. Let f be a hyperbolic transcendental entire function. Then the following are equivalent: (a) all Fatou components of f are bounded, and (b) f has no asymptotic value, and every Fatou component contains only finitely many critical points.
  2. If (b) in 1. holds, and additionally the number of critical points (counting multiplicity) belonging to the same Fatou component is uniformly bounded, then the Julia set is locally connected.

In the case where f has no asymptotic values and precisely two critical values, we obtain a particularly strong dichotomy. Indeed, in this case either all Fatou components are bounded quasidiscs, or all Fatou components are unbounded, with non-locally connected boundaries. This result appears to be new even for the (full) sine/cosine family.

We note that, together with a recent construction of Chris Bishop, our result implies that hyperbolic entire functions may have locally connected Julia sets, but still have very complicated dynamics ‘near infinity’.

We also give examples to show that our results are optimal:
a) We construct examples of entire functions with three critical values, no asymptotic values, and both bounded and unbounded Fatou components;
b) We construct an example of an entire function with two critical values such that all Fatou components are bounded quasidiscs, but the Julia set is not locally connected.

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