Period transfer from a quadratic map to its Lorenz sign lift ($\mathrm{sgn}((g^n{)}^{\prime })=\pm 1⟹\text{least period }n\text{ or }2n$)

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