Transcendental Julia sets of minimal Hausdorff dimension

We show the existence of transcendental entire functions f: C → 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 [Eremenko points and the structure of the escaping set. Trans. Amer. Math. Soc. 372 (5) (2019), 3083–3111]. Our proof relies on a quasiconformal-surgery approach developed by Burkart and Lazebnik [Interpolation of power mappings. Rev. Mat. Iberoam. 39 (3) (2023), 1181–1200].