Fine properties of LpLp-cocycles which allow abundance of simple and trivial spectrum

In this paper we prove that the class of accessible and saddle-conservative cocycles (a wide class which includes cocycles evolving in GL(d,R)GL(d,R), SL(d,R)SL(d,R) and Sp(d,R)Sp(d,R)) LpLp-densely have a simple spectrum. We also prove that for an LpLp-residual subset of accessible cocycles we have a one-point spectrum. Finally, we show that the linear differential system versions of previous results also hold and give some applications.