= Solution
After one iteration $y_n\in\{1,-1\}$, so work on that invariant sign sheet. For $x_n\ne0$,
$$
\begin{aligned}
z_{n+1}&=-x_{n+1}y_{n+1}
=\mu-A\operatorname{sgn}(y_n)\operatorname{sgn}(x_n)|x_n|^\delta\\
&=\boxed{\mu+A\operatorname{sgn}(z_n)|z_n|^\delta},
\end{aligned}
$$
since $z_n=-x_ny_n$ and $|y_n|=1$. This is the <signed gluing-map reduction>. If an arbitrary initial value of $y$ is admitted, the formula applies after that first iterate, not before it. The point $z=0$ represents a <separatrix> at which the original flow return is singular, even though the scalar map can be extended continuously there.
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