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:

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          [![z^3 & lambdas][1]][1]
          ^{L-to-R:    $\lambda=0.2-0.1 i$;   $\lambda=2-i$;    $\lambda=0.07-0.5i$.}

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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 $k=5$, we should see nonagonal symmetries?

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:

---

          [![z^3 & lambdas][1]][1]
          ^{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?