Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 55 / 4 / i / Solution

The Feigenbaum period-doubling map is quadratic-critical at zero: $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Differentiating the renormalization equation once gives $g^{\prime }(x)=g^{\prime }(g(\beta x))g^{\prime }(\beta x)$. Differentiate again and set $x=0$:Cancellation of the nonzero second derivative therefore givesThe quadratic critical-point property matters. The normalization and functional equation alone, without specifying the Feigenbaum solution class, would not justify cancelling $g^{\prime \prime }(0)$.