# Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the Julia sets of the map $$z \mapsto z^n + \lambda / z^k$$ with $$z,\lambda \in \mathbb{C}$$, and the exponents $$n,k \in \mathbb{N}$$. For example, for $$n=k=3$$, here are three Julia sets depictions for three different (relatively small) $$\lambda$$'s: L-to-R:    $$\lambda=0.2-0.1 i$$;   $$\lambda=2-i$$;    $$\lambda=0.07-0.5i$$.
There is obvious hexagonal symmetry independent of $$\lambda$$. My question is:

Q. Given $$n$$ and $$k$$, are the symmetries of the Julia sets of this map, for relatively small $$\lambda$$, known? Should they always have $$(n+k)$$-gon-like symmetries?

I can imagine that many details of these maps are unknown, but perhaps at this high-level viewpoint, the gross structure of the Julia sets is known? So for $$n=4$$ and $$k=5$$, should we expect to see nonagonal symmetries?

Yes. Let $$\zeta=e^{2\pi i/(k+n)}$$. Then $$f(\zeta^j z)=\zeta^{nj}f(z)$$. Iterating this, the orbit of $$\zeta^j z$$ agrees with the orbit of $$z$$ up to multiplying by powers of $$\zeta$$. In particular, $$z$$ lies in the Julia set if and only if $$\zeta z$$ lies in the Julia set.
• Thank you! ${}$ Mar 16 '19 at 13:09