Consider the function family given by $f_\lambda(z) = z - p_\lambda(z)/p_\lambda'(z)$ where $p_\lambda(z) = (z^2 - 1)(z - \lambda)$. Every attracting cycle and every rational neutral cycle of $f_\lambda$ attracts the one critical point of $f_\lambda$, which is $\lambda/3$ (see this other MathOverflow question and Alexandre Eremenko's answer for context).

If the sequence of iterates, $f^n_\lambda(\lambda/3)$, converges to a fixed point of $f_\lambda$ then the fixed point is one of the roots of $p_\lambda$, that is $1$, $-1$, or $\lambda$. If we mark the points $\lambda$ in $\mathbb{C}$ such that $f^n_\lambda(\lambda/3)$ converges to each of these fixed points then the result is an image that is reminiscent of a Newton basin fractal. Let $K$ be the set of values $\lambda\in\mathbb{C}$ such that $f^n_\lambda(\lambda/3)$ converges to a fixed point of $f_\lambda$. My question is:

Is the border, $\partial K$, a Julia set, and if so then for what function family $g$ does $J(g) = \partial K$?

I suspect that the answer is "yes" and that $g$ is a family of iterates of a rational function. In the attached images, the very bright green points are the ones nearest to the border $\partial K$.

A note about the little Mandelbrot sets visible in two of the attached images: In these images the white points indicate parameters $\lambda$ such that $f^n_\lambda(\lambda/3)$ converges to a rational neutral cycle of period greater than $1$. The white points indicate the bifurcation locus of $f_\lambda$ ($J(f_\lambda)$ is not continuously determined, in the sense of the Hausdorff metric, by the parameter at these points). The white points appear to be composed of many scaled and rotated copies of the boundary of the Mandelbrot set and I believe they are contained by $\partial K$. Perhaps that is a useful thing to know when searching for $g$.

nottrue that your set is the Julia set of a rational function. (If you care enough, you should be able to fashion a formal proof, e.g. from the fact that the complement contains some domains bounded by analytic curves, in the small Mandelbrot copies.) On the other hand, a one-dimensional bifurcation locus can usually be described as the locus of non-normality of a suitable family of analytic functions, formed by taking the iterates of the free critical point. $\endgroup$ – Lasse Rempe-Gillen Dec 3 '12 at 22:47