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

Abstract

Let $AC_D(M,SL(2,\mathbb R))$ denote the pairs $(f,A)$ so that $f\in \mathcal A\subset \text{Diff}^{1}(M)$ is a $C^{1}$-Anosov transitive diffeomorphisms and $A$ is an $SL(2,\mathbb R)$ cocycle dominated with respect to $f$. We prove that open and densely in $AC_D(M,SL(2,\mathbb R))$ (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 $\mathcal{R}\subset \text{Aut}_{Leb}(M)\times L^\infty(M,SL(2,\mathbb R))$, 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 $\text{Aut}_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$.