Marginally unstable discrete-time linear switched systems with highly irregular trajectory growth

We investigate the uniform stability properties of discrete-time linear switched systems subject to arbitrary switching, focusing on “marginally unstable” regime in which system is not Lyapunov stable but trajectories cannot escape infinity at exponential speed. For a discrete this type without switching fastest-growing trajectory must grow as an exact polynomial function time, and significant body prior research has focused investigating how far intuitive picture can be extended from cases where present. In note we give example family three dimensions, with two states, for intuition fails badly: generic member maximal rate growth escaping made arbitrarily slow along one subsequence times yet also faster than any prescribed slower-than-linear complementary times. Using construction new counterexamples conjecture Chitour, Mason Sigalotti obtain negative answer related question Jungers, Protasov Blondel. Our examples have additional feature that marginal instability are densely intermingled same parameter space.