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

Transcendental Julia Sets of Minimal Hausdorff Dimension