Connected and locally connected filled Julia sets

Question

For any polynomial map $f$ we can define the filled Julia $K$ to be closure of the complement of $ \Omega$ in $\mathbb{C}$ of the basin of infinity $$\Omega = \{z \in \mathbb{C}; f^{\circ n}(z)\rightarrow \infty \}.$$

Thus the boundary of $K$ is what is often called the Julia set. Now there are examples like $f=z^2$ for which the filled Julia set is connected and locally connected. My question is roughly speaking: How many such polynomial maps do exist? Or more precisely. Are there infinitely many non-equivalent such polynomial maps in any degree? By equivalence I mean the usual one. Two endomorphisms $f,g$ of the projective line are equivalent if there exists a Möbius transformation $\mu$ such that $\mu\circ f\circ \mu^{-1} =g$. If the second question has a positive answer I would be very interested in whether the proof is constructive. Any comment or reference would be greatly appreciated.

Answer 1

There infinitely many non-equivalent such polynomial maps of even degree:

Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an even integer and $c$ real. Then the Julia set of $f$ is either totally disconnected or locally connected.

Genadi Levin and Sebastian Van Strien. "Local Connectivity of the Julia Set of Real Polynomials", Annals of Mathematics, 147, no. 3 (1998): 471—541. doi:10.2307/120958