Why Study Critical Polynomials?

In dynamical systems, I often read about the post-critical orbits.
As in take a moduli space of functions $f$ which are self maps. Find general critical points, and see where they orbit. They would then be polynomials in some variables if we allow $f$ to be parameterised. Those are called critical polynomials. It could go by another name also, but I’m unsure.

For instance the moduli space of quadratic polynomials can be a parameterised as $f = z^2 + c : \mathbb{C} \cup \{\infty\} \to \mathbb{C} \cup \{\infty\}$, where $c \in \mathbb{C}$. Then $f$ has critical points $0$ and $\infty$, the latter being fixed.
Iterations of $0$ would give us the critical polynomials.

$0,\, c,\, c^2 + c, (c^2 + c)^2 + c\dots$

I know that they are important in dynamical systems, and their roots have some importance. But I don’t know why they are important at all.

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Critical polynomials are important because critical orbits are important. Critical orbits are important because they largely determine the possible global dynamics of a polynomial. Here are some well known results in complex dynamics to illustrate this:

Lemma:

An orbit is super-attractive if and only if it contains a critical point.

Theorem 1:

Suppose that $f$ is a complex polynomial with an attractive, super-attractive, or neutral orbit. Then the basin of attraction of that orbit must contain a critical point.

Corollary:

The number of attractive, super-attractive, and neutral orbits cannot exceed the number of critical points.

Theorem 2:

If all the critical orbits of a polynomial diverge to $\infty$, then Julia set of that polynomial is totally disconnected

Corollary:

If the obit of zero under iteration of $f_c(z)=z^2+c$ diverges to $\infty$, then the Julia set of $f_c$ is totally disconnected.


Theorem 2 and its corollary show why we care about whether the critical orbit escapes or not. As an application of the lemma, consider the problem of finding the so-called “hyperbolic components” of the Mandelbrot set. These are just the different, conspicuous pieces of the Mandelbrot set that we see – like the cardioids and circles. They all contain exactly one $c$ value with the property that the corresponding function $f_c$ has a super-attractive fixed point. Thus, we can find these by finding $c$ values where zero maps back to itself under iteration of $f_c$. Put another way, these are the roots of the critical polynomials associated with $f_c$.

This is illustrated in the following figure:

enter image description here

The green and red dots are all roots of some critical polynomial of $f_c$. If you’re curious about the yellow dots, check out this answer, which is where the picture comes from.

As another illustration, have a look at this image of the bifurcation diagram for the quadratic family, with plots of the critical polynomials super-imposed on top:

enter image description here

Note that the curves mesh quite nicely with the image; the points naturally cluster near those curves. The curves are evident, even when they aren’t explicitly drawn. Also, the periodic windows arise at the roots of the critical polynomials because, again, that’s where a super-attractive orbit can arise.