# Semi-hyperbolic rational maps and size of Fatou components

## TL;DR

This paper extends a theorem relating the size of Fatou components to the Hausdorff dimension from Sierpiński carpet Julia sets to more general semi-hyperbolic rational maps, confirming a conjecture by Merenkov and Sabitova.

## Contribution

It generalizes previous results to semi-hyperbolic rational maps and proves a stronger version of the conjectured relationship between Fatou component sizes and Julia set dimension.

## Key findings

- Extended the theorem to semi-hyperbolic rational maps.
- Proved a stronger version of the conjecture by Merenkov and Sabitova.
- Related peripheral circle diameters to Hausdorff dimension in broader settings.

## 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.