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Register a new userSimplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems
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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.
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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 hyperbolic systems, such as shift maps or Axiom A diffeomorphisms. In this paper we propose to extend such criteria to situations where the base map is just partially hyperbolic. This raises several new issues concerning, among others, the recurrence of the holonomy maps and the (lack of) continuity of the Rokhlin disintegrations of $u$-states. Our main results are stated for certain partially hyperbolic skew-products whose iterates have bounded derivatives along center leaves. They allow us, in particular, to exhibit non-trivial examples of stable simplicity in the partially hyperbolic setting.
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.
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 shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.
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.
We prove that the set of fiber-bunched $SL(2,mathbb{R})$-valued H{o}lder cocycles with nonvanishing Lyapunov exponents over a volume preserving, accessible and center-bunched partially hyperbolic diffeomorphism is open. Moreover, we present an example showing that this is no longer true if we do not assume acessibility in the base dynamics.
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