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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 of cocycles and thus extends the previous result of the second author and Potrie [KP16] to higher dimensions.
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 $alpha$
Let $G$ be a semisimple Lie group acting on a space $X$, let $mu$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated s
The $pi_2$-diffeomorphism finiteness result (cite{FR1,2}, cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolu
Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of $C^r$, $0leq r leq infty$ (or analytic) $k+1$-tuples of quasi periodic coc
We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.