Quadratic polynomial dynamics

Remarks

Every polynomial $p(z) = a z^2 + b z + d$ with $a \ne 0$ is conjugate to exactly one $f_c$ by an affine map $z \mapsto \alpha z + \beta$. This is why one parameter suffices. The only critical point of $f_c$ in $\C$ is $0$. The set $K_c$ is connected if and only if the orbit of $0$ is bounded, and otherwise $K_c$ is a Cantor set. Example: for $c = 0$, $f_0^n(z) = z^{2^n}$ is bounded exactly when $\abs{z} \le 1$, so $K_0$ is the closed unit disc and $J_0$ is the unit circle. Non-example: $z \mapsto z^3 + c$ is not in the quadratic family, since its degree is $3$.