= Leading polynomial approximation to a renormalization fixed point
Suppose $g(x)=1+c x^d+O(x^{d+2})$, with even $d$ and $c<0$. Expanding the <period-doubling renormalization operator> exactly at zero gives $a^{d-1}g'(1)=1$. Approximating globally by $1+c x^d$ gives $c\approx a-1$ and $g'(1)\approx dc$, hence $d(a-1)a^{d-1}\approx1$. For $d=2$, $a\approx(1-\sqrt3)/2$, $g(x)\approx1-(1+\sqrt3)x^2/2$ and $\alpha\approx1+\sqrt3$. The coefficient relation is a polynomial-truncation approximation, not an exact identity for the full analytic <fixed point>.
// Target: numerical-analysis.bigb
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