Dynamics of generic multidimensional linear differential systems

We prove that there exists a residual subset R (with respect to the C-0 topology) of d-dimensional linear differential systems based in a mu-invariant flow and with transition matrix evolving in GL(d, R) such that if A is an element of R, then, for mu-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Lyapunov spectrum is trivial. Moreover, in the conservative setting, we obtain the dichotomy: dominated splitting versus zero Lyapunov exponents.