 Sign lift of an even map

If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1{)}^n{g}^n(x){\prod }_{k=0}^{n-1}\mathrm{sgn}(g^k(x))$. The step uses oddness of $f$ on nonzero arguments. A one-sided assigned value of $f(0)$ can invalidate that step after a critical hit; for $g(x)=\mu -x^2$, $\mu >0$, $x=\sqrt{\mu }$, the claimed second-iterate formula would give $+\mu$ while the actual value is $-\mu$.

// Target: dynamical-systems.bigb