Density of positive Lyapunov exponents for symplectic cocycles

Abstract

We prove that Sp$(2d,\mathbb{R})$-cocycles, HSp($2d$)-cocycles and pseudo-unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non periodic dynamical systems. For Schrödinger operator on the strip, we prove a similar result for density of positive Lyapunov exponents. This generalizes a result of A. Avila [2] to higher dimensions.