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I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the iteration of these functions. It is apparent that $f_a^{\circ n}(x)$ should behave like $\text{b}^{2^n}$, but calculating the value of the base $b$ in terms of $a$ and $x$ has been more difficult than I anticipated. We may solve for $b$ using a limit: $$b_a(x)=\lim_{n\to\infty} \big[f_a^{\circ n}(x)\big]^{2^{-n}}$$ but this is as far as I have gotten, and I can’t make this any neater. The best thing I have managed to find is the rather trivial observation $$b_a(x^2+a^2)=b_a^2(x)$$ and I have noticed that the graph of $b_a(x)$ typically looks like (but is not) a hyperbola.

Can anyone calculate any special values of $b_a(x)$ in closed form, or in terms of an infinite series or integral? Can you find any more interesting (non-trivial) functional or differential equations?

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  • $\begingroup$ To get some idea of how complicated this can get, see oeis.org/A000058 and the links therein, for $f(x)=x^2-x+1$. Or oeis.org/A003095 for $f(x)=x^2+1$. $\endgroup$ – Gerry Myerson Mar 2 at 4:03
  • $\begingroup$ Had a look at those sites? Convinced? $\endgroup$ – Gerry Myerson Mar 3 at 5:33
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From Wikipedia, it states that the $n$th iteration of $$ax^2+bx+\frac{b^2-2b-8}{4a}$$ is $$\frac{2c^{2^n}+2c^{-2^n}-b}{2a}$$ where $$c=\frac{2ax+b\pm\sqrt{(2ax+b)^2-16}}{4}$$


Thus, we can solve the case $f(x)=x^2-2$.

$$b_{\pm 2i}(x)=\frac{x+\sqrt{x^2-4}}2$$

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  • $\begingroup$ Yes, but that's a very special case. In the general quadratic case, there's no closed-form formula. $\endgroup$ – Gerry Myerson Mar 2 at 4:01

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