Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 53 / 2 / d

Write the local Taylor series at the critical point aswhere $d$ is even, and put $a=g(1)$. The fixed-point equation first gives $g(g(0))=g(1)=a$, consistently with normalization. Expanding the inner function at zero and the outer one at one,The quadratic remainder in the outer expansion is of order $x^{2d}$, at least $x^{d+2}$ because $d\ge 2$. Comparing the nonzero $x^d$ coefficients proves the exact identityTo obtain the requested first-order relation, use the leading polynomial approximation to a renormalization fixed point: retain only $g(x)\approx 1+c{x}^d$. Then $a\approx 1+c$, while $g^{\prime }(1)\approx dc$, giving the truncated equationThe equality belongs to this truncation; higher analytic coefficients generally change $g^{\prime }(1)$ and $a$, so it is not an additional exact equation for the full fixed point.

For $d=2$, solve $2a(a-1)=1$. The root compatible with $-1<a<0$ is $a=(1-\sqrt{3})/2$; the other root is inadmissible. ThusThis is a coarse leading approximation, not the more accurate universal value $2.5029\dots$ obtained after retaining higher coefficients.