= Period-doubling renormalization operator
{title2=$\mathcal T(f)(x)=f(f(ax))/a$}
= Doubling operator
{synonym}
= Period-doubling renormalization
{synonym}
For a normalized even <unimodal interval map> with $f(0)=1$ and $a=f(1)\in(-1,0)$, define
$$
\mathcal T(f)(x)=a^{-1}f(f(ax)).
$$
This returns the second iterate near the <critical point> to the original spatial and height normalization. Then $\mathcal T(f)(0)=f(1)/a=1$. A restrictive-interval condition is also needed for the composition to remain unimodal and map $[-1,1]$ into itself; normalization alone is not sufficient.
// Target: dynamical-systems.bigb
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