Were any to exist, would "non-hyperbolic" Mandelbrot components be detectable visually?

Question

I have heard of this conjecture that there exists no "non-hyperbolic" components in the Mandelbrot set, which is believed but not proven. My question is, though, what would such a component "look" like if it existed? (E.g. are there other maps than $z \mapsto z^2 + c$ which have them?) Were, hypothetically, this conjecture to be false, could someone playing around with fractal zoom programs theoretically stumble upon a "something" that would be immediately recognizable and were a picture of it with coordinates published, would falsify the conjecture? Or would they be too subtle to detect in that fashion? But if they could be, what would our hypothetical zoomer encounter that would signal such a component? Likewise, does the proliferation of zoom footage which never features that, constitute "probabilistic" evidence in favor of this hypothesis that no such components exist?

Answer 1

First off, there are indeed other parametrized families of functions with the property that the corresponding bifurcation locus has non-hyperbolic components. One such example is the family of rational maps defined by $$R_a(z) = \frac{z^3-z}{-z^2+az+1}.$$ This family is studied in the paper "Dynamics and bifurcations of a family of rational maps with parabolic fixed points" by Jane Hawkins and Rika Hagihara. You can download a copy of the paper from Hawkins' website.

Here is an image of the parameter space taken from the paper:

It's a pretty simple matter to compute the derivative, of course: $$R_a'(z) = \frac{-z^4 + 2 a z^3+2 z^2-1}{\left(a z-z^2+1\right)^2}.$$ From here, it's easy to see that zero is always a neutral fixed point, since $R_a(0)=0$ and $R_a'(0)=-1$. As it turns out, the point at $\infty$ is always a neutral fixed point at well. Of course, if either of those points has a non-empty basin of attraction, then a critical point must lie in that basin. Since the numerator of the derivative is a fourth degree polynomial, there are always four critical points and to understand the parameter space, we need to iterate from each of the four points and classify the parameters according to the various possibilities that occur. The image from the paper is colored according to the following classification:

• The darker gray region that lies mostly above the red curve is a hyperbolic component, ie. at least one critical point converged to an attractive orbit.
• In the lighter gray region, two critical points are attracted to zero and two to $\infty$.
• In the white region, three critical points are attracted to zero and one to $\infty$.

Thus, the light gray and white regions are non-hyperbolic. I do think they look a bit non-Mandelbrot like but I don't know of a quantitative way to state this.

Having said all this, I think it's pretty clear why this particular behavior can't happen in the Mandelbrot set - with only one critical point, there can be at most one neutral periodic orbit and there's no way for the set of critical points to bifurcate in its choices of which neutral orbits to converge to.

Also, I do think that there is other probabilistic evidence against the conjecture. The simplest is to compute the area from the outside using an escape time approach and subtract the area from the inside using orbit detection. I don't know the results off of the top of my head, though.