The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in $${\fancyscript G}$$ IC (X, SL(2,ℝ))

Abstract

Let (X, \({\mathfrak{S}}\) (X), \({\mathfrak{m}})\) be a probability space with σ-algebra , \({\mathfrak{S}}\)(X) and probability measure \({\mathfrak{m}}.\) The set V in \({\mathfrak{S}}\)is called P-admissible, provided that for any positive integer n and positive-measure set Vn ∈ \({\mathfrak{S}}\) contained in V , there exists a Zn ∈ \({\mathfrak{S}}\) such that Zn ⊂ Vn and 0 < \({\mathfrak{m}}\)(Zn) < 1/n. Let T be an ergodic automorphism of (X, \({\mathfrak{S}})\) preserving \({\mathfrak{m}},\) and A belong to the space of linear measurable symplectic cocycles