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

We prove that if a Hölder continuous fiber-bunched cocycle $\widehat{A}$ over an invertible hyperbolic transitive shift $\widehat{\mathrm{\Sigma }}$ satisfies an appropriate strong irreducibility condition on Grassmannians, then $\widehat{\mathrm{\Sigma }}$ admits an ergodic measure $\widehat{\mu }$ 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.