Transcendental Julia Sets of Minimal Hausdorff Dimension

Abstract

We show the existence of transcendental entire functions $f:\mathbb{C}\to \mathbb{C}$ with Hausdorff-dimension $1$ Julia sets, such that every Fatou component of $f$ has infinite inner connectivity. We also show that there exist singleton complementary components of any Fatou component of $f$, answering a question of Rippon and Stallard (arXiv:1703.11001). Our proof relies on a quasiconformal-surgery approach developed in arXiv:2101.04219.

References in corpus (7)

• Multiply connected wandering domains of entire functions
• An entire function with simply and multiply connected wandering domains
• On the uniform perfectness of the boundary of multiply connected wandering domains
• On multiply connected wandering domains of meromorphic functions
• Multiply connected wandering domains of meromorphic functions: internal dynamics and connectivity
• On connected preimages of simply-connected domains under entire functions
• A generalized family of transcendental functions with one dimensional Julia sets

Cited by in corpus (1)

• The escaping set in transcendental dynamics