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

TL;DR

This paper studies the spectral properties of $SL(2,\mathbb{R})$ cocycles over Anosov diffeomorphisms, showing that simple spectrum is generic while trivial or hyperbolic spectrum is prevalent in certain settings.

Contribution

It establishes generic simplicity of spectrum for dominated cocycles over Anosov maps and shows the prevalence of trivial or hyperbolic spectrum in broader contexts.

Findings

Open and dense pairs have simple spectrum with respect to the maximal entropy measure.

Residual subsets contain cocycles with trivial or hyperbolic spectrum.

Trivial spectrum is prevalent near area-preserving maps and in generic cocycles.

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 $\text{Aut}_{Leb}(M)\times L^p(M,SL(2,\mathbb R))$.