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