We prove that if a H\"older continuous fiber-bunched cocycle $\hat{ A}$ over an invertible hyperbolic transitive shift $\hat{ \Sigma}$ satisfies an appropriate strong irreducibility condition on Grassmannians, then $\hat{ \Sigma}$ admits an ergodic measure $\hat{ \mu}$ 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.
Categories: math.DS