# Symmetries for Julia sets of rational maps

## TL;DR

This paper explores the symmetries of Julia sets for rational maps, extending previous results on polynomial cases and discussing their implications for more complex rational functions and singular perturbations.

## Contribution

It provides partial extensions to Beardon's results on Julia set symmetries for rational maps and discusses their relevance to singularly perturbed maps.

## Key findings

- Extended symmetry group results to certain rational maps
- Highlighted increased complexity in rational case compared to polynomial case
- Discussed implications for singularly perturbed maps

## Abstract

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. Here, we offer partial extensions to Beardon's results on the symmetry group of Julia sets, and discuss them in the context of singularly perturbed maps.