**MSC:**- 93B05 Controllability
- 34D08 Lyapunov exponents

We study structural properties of linear time-varying discrete-time

systems. At first an associated system on projective space is

introduced as a basic tool to understand the linear dynamics. We

study controllability properties of this system, and characterize in

particular the control sets and their cores. Sufficient conditions

for an upper bound on the number of control sets with nonempty

interior are given. Furthermore exponential growth rates of the

linear system are studied. Using finite time controllability

properties in the cores of control sets the Floquet spectrum of the

linear system may be described. In particular, the closure of the

Floquet spectrum is contained in the Lyapunov spectrum.