Coexistence of measures with simple Lyapunov spectrum for fiber-bunched cocycles

We prove that if a Hölder continuous fiber-bunched cocycle A ^ Â ModifyingAbove upper A With caret over an invertible hyperbolic transitive shift Σ ^ Σ ̂ ModifyingAbove normal upper Sigma With caret satisfies an appropriate strong irreducibility condition on Grassmannians, then Σ ^ Σ ̂ ModifyingAbove normal upper Sigma With caret admits an ergodic measure μ ^ μ ̂ ModifyingAbove mu With caret with full support and product structure with simple Lyapunov spectrum if and only if any other ergodic measure with full support and product structure also has simple Lyapunov spectrum.