 Solution

For a real determinant-one map, bounded generic repeated transport requires eigenvalues $e^{\pm i\mu }$, equivalently $|\mathrm{tr}M|<2$. Here both planes have $\mathrm{tr}M=2-L_1{L}_2/f^2$, soFor positive drift lengths and $f>0$, this means $f>\sqrt{L_1{L}_2}/2$. The equalities are marginal and generally produce a Jordan-block growth rather than stable motion. Moving the reference point changes the map by a similarity transformation, $M\to PM{P}^{-1}$, so the eigenvalues and full-cell phase advance are unchanged.

Write the stable map using the Courant-Snyder parameters, also called Twiss parameters:At the chosen reference point $\beta =M_{12}/\sin \mu$, $\alpha =(M_{11}-M_{22})/(2\sin \mu )$, and ${\gamma }_T=-M_{21}/\sin \mu$. Repeated traversal givesIf the full ring contains $N_c$ identical cells, its one-turn phase is $N_c\mu$ modulo $2\pi$; $n$ ring turns use $n{N}_c\mu$. The displayed $M^n$ is for $n$ repetitions of the specified cell.