Renormalization of the tent map near its fixed point ($h\circ F_{\mu }^2\circ h^{-1}=F_{{\mu }^2}$)

For $1<\mu \le \sqrt{2}$, the interval $J=[1/(\mu +1),\mu /(\mu +1)]$ is invariant under the second iterate of the tent map. The affine coordinate $h(x)=(\mu /(\mu +1)-x)/((\mu -1)/(\mu +1))$ conjugates that restriction to the tent map with parameter ${\mu }^2$. Its two linear pieces give the identity directly. This turns the two-iterate horseshoe threshold into a four-iterate threshold and explains why chaotic invariant sets need not have periodic points near the original nonzero fixed point.

For $1<s\le \sqrt{2}$, 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,Successive squaring eventually brings every $s>1$ into $(\sqrt{2},2]$. 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