 Renormalization of the tent map near its fixed point

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.