Consider $f(x)=\dfrac{ax+b}{cx+d}$, where $c\neq0$ and $f(x)$ is not equal to a constant. Is it necessarily true that $f^{[n]}(x)=f(x)$ for some natural number $n > 1$?
Iterations of $f(x)=\dfrac{ax+b}{cx+d}$
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functional-equations
function-and-relation-composition
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0you want to find a,b,c,d such that there exists n such that for all x $f^{(n)}=f$? (where $f^{2)}(x)=f(f(x))$ – 2012-11-19
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0@Amr: as I interpret the question (looking at the three uses of the word "must") the question is whether _all_ $f$ have the property that such an $n > 1$ exists. – 2012-11-19
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0I think that he might not want your counterexamples because he does not want c to be zero – 2012-11-19
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0@Amr: ah, I didn't catch that. – 2012-11-19