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 V _n ∈ ${\mathfrak{S}}$ contained in V , there exists a Z _n ∈ ${\mathfrak{S}}$ such that Z _n ⊂ V _n and 0 < ${\mathfrak{m}}$(Z _n ) < 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

${\fancyscript G}$ _IC (X, SL(2,ℝ)) := {A : X → SL(2,ℝ)| log ∥A ^±1(x)∥ ∈ ${\Bbb L}$ ¹(X, ${\mathfrak{m}}).$

We prove that for any P–admissible set V and ε > 0, there exists a B ∈ ${\fancyscript G}$ _IC (X, SL(2,ℝ)) such that

max{∥A(x) − B(x)∥, ∥A ⁻¹(x) − B ⁻¹(x)∥} < ε for all x ∈ V, A(x) = B(x) for all x ∈ X \ V,

and B has the simple Lyapunov spectrum over the system (X, ${\mathfrak{m}},$ T).