Find all functions that satisfy $$ f (f (x))=x^2-3x+4$$ Any thoughts and approachs to find $f (x)$ ?
closed as off-topic by Travis Willse, abiessu, GNUSupporter 8964民主女神 地下教會, Dietrich Burde, Namaste Jan 19 '18 at 0:41
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There is an $f$ that is $C^\infty$ for $x > 3/2.$ It is real analytic for $x \neq 2.$ You can get $C^1$ on the whole line by reflecting across $x = 3/2.$
The method is difficult, see similar http://math.stackexchange.com/questions/208996/half-iterate-of-x2c/209653#209653 for $x^2 + x.$ This version seems more complete How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? My best answer on this stuff is http://mathoverflow.net/questions/45608/formal-power-series-convergence/46765#46765 on $f(f(x)) = \sin x,$ and Gottfried posted a lovely picture of the resulting $f.$
We know that we need to use the intricate Ecalle method because the function has a fixed point at $x=2,$ with slope equal to one there. The curve is tangent to the line $y=x,$ which rules out easier methods.