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We prove a Livv{s}ic-type theorem for Holder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,mu)$ is a non-uniformly hyperbolic system and $A:M to GL(d,mathbb{R}) $ is an $alpha$-H{o}lder continuous map satisfying $ A(f^{n-1}(p))ldots A(p)=text{Id}$ for every $pin text{Fix}(f^n)$ and $nin mathbb{N}$, there exists a measurable map $P:Mto GL(d,mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $mu$-almost every $xin M$. Moreover, we prove that whenever the measure $mu$ has local product structure the transfer map $P$ is $alpha$-H{o}lder continuous in sets with arbitrary large measure.
We prove that generic fiber-bunched and Holder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a u-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched cocycles.
In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.
Criteria for the simplicity of the Lyapunov spectra of linear cocycles have been found by Furstenberg, Guivarch-Raugi, Goldsheid-Margulis and, more recently, Bonatti-Viana and Avila-Viana. In all the cases, the authors consider cocycles over hyperbol
We prove the so called Livv{s}ic theorem for cocycles taking values in the group of $C^{1+beta}-diffeomorphisms of any closed manifold of arbitrary dimension. Since no localization hypothesis is assumed, this result is completely global in the space
This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural g