Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 85 / 2 / c / Solution

For a period-$n$ orbit of $g(x)=\mu -x^2$ that avoids zero, the chain rule gives its multiplier of a periodic orbit of an iteration asThe sign of $g^{\prime }$ is positive on the negative half-axis and negative on the positive half-axis, exactly as the parabola sketch shows. ConsequentlyThis product is independent of the chosen starting point because changing $\hat{x}$ only cyclically reorders the factors. If the orbit contains zero, the derivative product is zero and its orientation is undefined. One must then use the actual zero periodic-orbit multiplier, rather than infer a positive or negative orientation from the special assignment $\mathrm{sgn}(0)=1$. The nonzero-periodic-orbit multiplier cases needed in part (d) avoid this exception automatically.