 Solution

After one iteration $y_n\in \{1,-1\}$, so work on that invariant sign sheet. For $x_n\ne 0$,since $z_n=-x_n{y}_n$ 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.