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Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is $$f_1(x)=f(x)$$ $$f_2(x)=f \circ f(x)$$ $$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$

Is there a way to find a formula for $f_n$?

I tried to write down $f_2$, $f_3,\ldots$, but I don't see any pattern.

Michael Hardy
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2 Answers2

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After a simple change of variables, you will be iterating $z^2+c$ for some $c$. There is a simple formula for the $n$th iterate when $c=0$ or $c=-2$. But not otherwise.

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Double-angle formula: $\cos 2\theta = 2\cos^2-1$, so if we write $z=2\cos\theta$, then we get $f_1 = z^2-2 = 2\cos 2\theta$ ; $f_2 = (z^2-2)^2-2 = 2\cos 4\theta$ ; and so on ... $f_n = 2\cos(2^n\theta)$. If you like, put $\theta = \arccos(z/2)$ into these.

GEdgar
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  • I can't figure the simple expression for the case $c=-2$. I tried

    $$f_1=z^2-2$$ $$f_2=z^4-4z+2$$ $$f_3=f_2^2-2$$

    But at $f_3$, I'm blocked, and I can't identify a formula. Thanks.

     –  Sep 22 '11 at 14:25
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    For the $c=-2$ case, remember the double-angle formula for cosines. – GEdgar Sep 22 '11 at 14:48
  • Sorry, I thought I had it clear, but I'm not sure of how to apply the double-angle formula here. Can you show me for $f_2$. Thanks. –  Sep 22 '11 at 21:35
  • Thank you very much GEdgar. It's was the first for me to use the angle formula is such context. I was very hesitant at each step. But your precision make me confident now. Thanks. –  Sep 23 '11 at 10:42
  • Is the formula we derived valid $\forall z \in \mathbb R$? Sorry if this is elementary, I'm confused since $0 \leq cos(\theta) \leq 1, \forall \theta \in \mathbb R$. –  Sep 23 '11 at 23:51
  • There is no reason $z$ must be real, either; it could be complex. So allow $\theta$ to be complex, too. However, if you want real $z$ with $z>2$ there is a similar formula with hyperbolic cosine. – GEdgar Sep 24 '11 at 00:57
  • @Nicolas: or you can recast the whole bit in terms of Chebyshev polynomials of the first kind... – J. M. ain't a mathematician Sep 29 '11 at 18:07
  • @J. M.: In other words, you are asking for a "formula" for the Chebyshev polynomials...??? – GEdgar Sep 29 '11 at 19:29
  • I sure as peas ain't. ;) I'm noting the cosine/hyperbolic cosine bit can be unified if you switch to Chebyshev's formulation, that's all... – J. M. ain't a mathematician Sep 29 '11 at 19:30
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I don't believe there is a "nice" formula for $f_n$ or even a pattern. Here's my reasoning:

If the graph of $y = a + bx^2$ intersects the line $y=x$, then there can be chaotic behavior in the values $f_n(x)$ for general $x$. See the neat animation on the Wiki article for "cobweb plot":

http://en.wikipedia.org/wiki/Cobweb_plot

Hope this helps!

Shaun Ault
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    Of course, there are some special cases of note. If $a = 0$, then it's fairly easy to find the general formula: $f_n(x) = b^{2^n - 1}x^{2^n}$. – Shaun Ault Sep 22 '11 at 01:16