[Newton basin fractals](https://en.wikipedia.org/wiki/Newton_basin) are visualizations of the Julia sets of functions of the form:


$$f\_p(z) = z - p(z)/p'(z)$$


where $p$ is a complex polynomial. My question is:



> 
> When is the Julia set, $J(f\_p)$, continuously determined by the roots of $p$, in the sense of the Hausdorff metric?
> 
> 
> 


I have convinced myself that it is a *necessary* condition that the roots of $p$ be simple, and I'm happy to elaborate on my reasoning, but I suspect that the statement is either obvious (or obviously *false*) to experts. (**Update:** It turns out that it is [obviously true](https://mathoverflow.net/a/223686/16518).)


B. Krauskopf and H. Kriete [proved that](http://iopscience.iop.org/0951-7715/9/2/018):



> 
> If a sequence of meromorphic functions $f\_n$ converges uniformly on compact sets to a limit function $f$, and $F(f)$ is a union of basins of attracting periodic orbits, and $\infty\in J(f)$, then $J(f\_n)$ converges to $J(f)$ in the Hausdorff metric.
> 
> 
> 


And at first glance that theorem looks like a great starting point for showing that the simple roots condition is *sufficient*. Unfortunately it is not the case that $F(f\_p)$ is a union of attracting basins whenever $p$ has simple roots.


Is anything more known about this? Perhaps there is some other condition (not merely simple roots) that makes $F(f\_p)$ a union of attracting basins.


I expected for this to be a well studied problem. An answer to the related question of "When is convergence of Newton-Raphson iteration not sensitive to perturbation of roots?" seems like it would have useful applications. But I have not found much beyond results like Krauskopf and Kriete's theorem, above, which don't *quite* apply.


**Edit:** To give the question a little more context and motivation: [Here is a program](http://aaronegolden.com/julia.html) to demonstrate that in *some* sense, perhaps not exactly the way I've phrased it, a Newton basin fractal may transform continuously under changes to the roots of its polynomial. In the demonstration ([which requires a WebGL enabled browser](http://www.khronos.org/webgl/wiki/Getting_a_WebGL_Implementation)) the Newton basin fractal for the polynomial $z\to (z^2 - 1)(z - \lambda)$ is rendered in a 10x10 box centered at the origin. Clicking and dragging updates $\lambda$ and the resulting basins.


**Edit 2:** I've updated [the demonstration](http://aaronegolden.com/julia.html) in an attempt to get more information onto the screen. Now the usual Newton basin rendering is blended with a rendering of the parameters $\lambda$ such that the sequence of iterates starting at the critical point converges very slowly. These are the parameters for which the Fatou set might include components that are not basins of attracting periodic orbits, preventing application of Krauskopf and Kriete's theorem.


The roots are highlighted in yellow and the critical point is highlighted in cyan. It's easy to see that the Julia set experiences a discontinuous transformation when any two roots overlap, and that when $\lambda$ is in the "problem area" the critical point is sort of pressed towards the Julia set (which makes sense given how the problematic parameters are determined in the first place). But even as I move $\lambda$ through the problematic region I don't see the Julia set transforming in a discontinuous way, certainly nothing as *obviously* discontinuous as the transformations that occur when two roots overlap. I wonder if the Julia set actually *is* continuously determined by $\lambda$ in this region, even if the theorem above isn't quite enough to prove it. 



When is a Newton basin fractal continuously determined by the roots of its polynomial?
The question of when the Julia set of a function depends continuously on the parameters is well-studied. For the case of polynomials, see Douady, *Does a Julia set depend continuously on the Polynomial?* Proceedings of Symposia in Applied Mathematics, 49, 91–135, 1994.


This implies that the answer for your question is negative. More precisely, consider the *bifurcation locus* of the family that you are looking at; i.e., the set of *unstable* parameter value. (It is well-known that such parameters exist; e.g. take points on the boundaries of little Mandelbrot set copies in the parameter plane.)


Now, in this locus, the set of parameters having a linearizable irrationally indifferent periodic point is dense. For these parameters, the periodic point is in the Fatou set, and hence has positive distance from the Julia set. On the other hand, arbitrarily small perturbations will lead to parameters with parabolic points, where the periodic point is in the Julia set. This proves that the Julia set does not depend continuously near such points.


(The answer is also negative near parabolic parameters.)


Of course, if you are in the complement of the bifurcation locus (say, with all critical points tending to attracting orbits), then the Julia set depends continuously, and even moves holomorphically.


There is always a *semicontinuous* dependence of the Julia set, due to the fact that repelling periodic orbits are dense, and that these are stable under perturbations. In other words, under small perturbations the Julia set can get *larger* in a discontinuous way, but not *smaller*. 


