= Period transfer from a quadratic map to its Lorenz sign lift
{title2=$\operatorname{sgn}((g^n)')=\pm1\quad\Longrightarrow\quad\text{least period }n\text{ or }2n$}
For a noncritical least-period-$n$ cycle of $g(x)=\mu-x^2$, the multiplier sign is $(-1)^n\prod_k\operatorname{sgn}(x_k)$. 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 $|(g^n)'|$ and $|(g^n)'|^2$. 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
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