A stability dichotomy for discrete-time linear switching systems in dimension two

We prove that for every discrete-time linear switching system in two complex variables and with finitely many switching states, either the system is Lyapunov stable or there exists a trajectory which escapes to infinity with at least linear speed. We also give a checkable algebraic criterion to distinguish these two cases. This dichotomy was previously known to hold for systems in two real variables, but is known to be false in higher dimensions and for systems with infinitely many switching states.