= Centered tent-map period-doubling renormalization
{title2=$h_s^{-1}T_s^2h_s=T_{s^2}$}
For $1<s\leq\sqrt2$, set $h_s(x)=-(s-1)x$ and $J_s=[-(s-1),s-1]$. The <tent map> sends $J_s$ to $[1+s-s^2,1]$, with disjoint interiors, and $T_s^2(J_s)\subseteq J_s$. Directly,
$$
h_s^{-1}\circ T_s^2\circ h_s=T_{s^2}.
$$
Successive squaring eventually brings every $s>1$ into $(\sqrt2,2]$. A <horseshoe for an interval map> of an iterate in the renormalized coordinate lifts to a horseshoe of an iterate of the original map.
// Target: dynamical-systems.bigb
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