  We introduce transfer matrices to describe the motion of particles in the
vicinity of the stable and unstable fixed points of longitudinal phase space
and use them to analyze the transfer of bunches between radio-frequency systems
operating at different harmonics and voltages. We then apply this formalism to
analyze the decoherence due to amplitude-dependent tune shift on the
asymptotically achievable longitudinal emittance and assess tolerances for the
injection.
