The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in G IC (X, SL(2,))

Let (X, S (X), m) be a probability space with .-algebra , S(X) and probability measure m. The set V in Sis called P-admissible, provided that for any positive integer n and positive-measure set V n . S contained in V , there exists a Z n . S such that Z n . V n and 0 < m(Z n ) < 1/n. Let T be an ergodic automorphism of (X, S) preserving m, and A belong to the space of linear measurable symplectic cocycles G IC (X, SL(2,.)) := {A : X . SL(2,.)| log .A ±1(x). . L 1(X, m) We prove that for any P.admissible set V and . > 0, there exists a B . G IC (X, SL(2,.)) such that max{.A(x) . B(x)., .A .1(x) . B .1(x).} < . for all x . V, A(x) = B(x) for all x . X \ V, and B has the simple Lyapunov spectrum over the system (X, m, T)