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

Abstract

We prove that open and densely in AC D (M, SL(d, R)), in appropriate topologies, the pair (f, A) has simple spectrum with respect to the unique maximal entropy measure µ f . Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in Aut Leb (M ) × L p (M, SL(d, R)).

Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems

“…Our main result here provides an affirmative answer to a conjecture of Viana in [34, p. 648], in the context of fiber-bunched cocycles: the set of linear cocycles whose Lyapunov exponents are all different contains an open and dense set of fiber-bunched cocycles. We remark that other similar questions have also attracted the attention of the community and have already been answered in some specific contexts (see [3,4,7,9,12,13,16,29,35] and the references therein). We also remark that for the dynamical cocycle, corresponding to the cocycle D f over the diffeomorphism f , fewer results are known.…”