Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics

Let $A{C}_D(M,SL(2,\mathbb{R}))$ denote the pairs $(f,A)$ so that $f\in \mathcal A\subset $\text{Diff}$^{1}(M)$isa$C^{1}$-Anosovtransitivediffeomorphismsand$A$isan$SL(2,\mathbb R)$cocycledominatedwithrespectto$f$. We prove that open and densely in $A{C}_D(M,SL(2,\mathbb{R}))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure ${\mu }_f$. On the other hand, there exists a residual subset $\mathcal{R}\subset {\text{Aut}}_{Leb}(M)\times L^{\mathrm{\infty }}(M,SL(2,\mathbb{R}))$, with respect to the separate topology, such that any element $(f,A)$ in $\mathcal{R}$ has trivial spectrum or it is hyperbolic. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in $$\text{Aut}$_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$.