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

Abstract

Let $A{C}_D(M,SL(2,\mathbb{R}))$ denote the pairs $(f,A)$ so that $f\in \mathcal{A}\subset {\text{Diff}}^1(M)$ 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 $A{C}_D(M,SL(2,\mathbb{R}))$ (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 $\mathcal{R}\subset {\text{Aut}}_{Leb}(M)\times L^{\infty }(M,SL(2,\mathbb{R}))$, 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 ${\text{Aut}}_{Leb}(M)\times L^p(M,SL(2,\mathbb{R}))$.

8 pages, revised version

References in corpus (4)

• Genericity of zero Lyapunov exponents
• $L^p$-generic cocycles have one-point Lyapunov spectrum
• Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles
• $C^0$-coerciveness of Moser's problem and smoothing area preserving homeomorphisms