Semi-hyperbolic rational maps and size of Fatou components

Abstract: Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${\'n}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${\'n}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.