  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.
