Sign lift of an even map ($f(x)=-\mathrm{sgn}(x)g(x)$)

If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1{)}^n{g}^n(x){\prod }_{k=0}^{n-1}\mathrm{sgn}(g^k(x))$. 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

For a noncritical least-period-$n$ cycle of $g(x)=\mu -x^2$, the multiplier sign is $(-1{)}^n{\prod }_k\mathrm{sgn}(x_k)$. A positive multiplier gives two symmetry-related period-$n$ cycles of the odd quadratic Lorenz map; a negative multiplier gives one period-$2n$ cycle. Their multipliers are respectively $|(g^n{)}^{\prime }|$ and $|(g^n{)}^{\prime }{|}^2$. Distinct points of a quadratic-map cycle cannot have the same absolute value, since the map is even and acts bijectively on its cycle; this proves the asserted least periods.

// Target: dynamical-systems.bigb