 Feigenbaum period-doubling map

The smooth quadratic-critical fixed point of period-doubling renormalization is normalized by $g(0)=1$ and $\beta =g(1)<0$. It has $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Differentiating its displayed functional equation twice at zero gives $g^{\prime }(1)=1/\beta$. Its restrictive intervals generate a Cantor attractor described by Feigenbaum geometric potential weights.