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I've finally resigned myself to believing that functions such as $f(x)=x^2+1$ have no iteration formulas in the closed form. I'm pretty sure that this is true, but I have no way of knowing for sure as of yet. Can anyone give me a way to determine and prove if a polynomial function can be iterated in the closed form?

I already know that functions of the form $f(x)=\frac{1}{a}(ax+c)^2-\frac{c}{a}$ can be iterated because they take the form $f(x)=(g\circ h\circ g^{-1})(x)$. Some other quadratics such as $f(x)=2x^2-1$ can be iterated using trigonometric functions. Maybe that helps.

Edit: Many quadratic functions "flip out" when you graph their iterates, and you get a strange and messy graph. These are usually the ones you can't iterate. The ones that can be iterated of the first form I mentioned have no inflection point, and those of the second type do, but each maximum and minimum has the same y-value. Just by messing around and graphing the iterates of a few parabolas with the line $y=x$ as a tangent at a single point, I noticed that their iterates also never have inflection points and don't behave erratically under iteration. It seems like they should be iteratable, but I can't figure out how.

Franklin Pezzuti Dyer
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  • Perhaps this previous question and answer helps. – Robert Israel Apr 25 '17 at 23:51
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    Don't know if this helps, but $f$ doesn't have any fixpoints in the reals. You're two examples both have fixpoint(s). – Χpẘ Apr 25 '17 at 23:53
  • A bit of information and possibly some links to further discussions/material can be find in the tetration-forum http://math.eretrandre.org/tetrationforum/showthread.php?tid=275 I did not dive in it today so it may or may not be relevant. But perhaps you can find more at the forum (I remember that the author once wanted to create a simple list for known cases of iterable elementary functions, but didn't find it without big effort) – Gottfried Helms Apr 27 '17 at 06:27

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As for quadratic polynomials, there are two special cases and only two, which have a finite iteration scheme:

1.) $\ ax^2+bx+ \frac{b^2-2b}{4a}$ iterates to $\frac{2A^{2^n} -b}{2a}$, where $A=\frac{2ax+b}{2}$;

2.) $ ax^2+bx+\frac{b^2-2b-8}{4a}$ iterates to $\frac {2B^{2^n}+2B^{-2^n}-b}{2a}$, where $B=\frac{2ax+b+\sqrt{(2ax+b)^2-16}}{4}$

Source: https://en.wikipedia.org/wiki/Iterated_function#Examples

As for higher-order polynomials, I cannot say. I have, however, found Carleman Matricies (https://en.wikipedia.org/wiki/Carleman_matrix) to be an interesting prospect provided you do not seek fractional iterations (that would require taking fractional powers of a matrix).

I hope you find this helpful.

TheForceApplied
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