= Sign lift of an even map
{title2=$f(x)=-\operatorname{sgn}(x)g(x)$}
If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1)^n g^n(x)\prod_{k=0}^{n-1}\operatorname{sgn}(g^k(x))$. 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
Back to article page
