Continuity of Lyapunov exponents is equivalent to continuity of Oseledets subspaces

Abstract

Abstract. We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.

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.…”

The set of fiber-bunched cocycles with nonvanishing Lyapunov exponents over a partially hyperbolic map is open

“…x be the Oseledets decomposition associated to A at the point x ∈ M , it follows from Proposition 3.1 of [4] that for any F A -invariant measure m, its conditional measures are of the form…”

Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms

“…If m k → m thenm k →m = H * m. Proof. Let ϕ : M × E → R be a continuous function, thenϕdm k = ϕ • H k dm k , ϕ • H is measurable but v → ϕ •H(x, v)is continuous for every x ∈ M , then by lemma A.3 we have that(9) ϕ • Hdm k → ϕdm.…”