Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 1825–1838 | Cite as

A Livšic Theorem for Matrix Cocycles Over Non-uniformly Hyperbolic Systems

  • Lucas Backes
  • Mauricio PolettiEmail author


We prove a Livšic-type theorem for Hölder 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 \(A:M \rightarrow GL(d,\mathbb {R}) \) is an \(\alpha \)-Hölder continuous map satisfying \( A(f^{n-1}(p))\ldots A(p)=\text {Id}\) for every \(p\in \text {Fix}(f^n)\) and \(n\in \mathbb {N}\), there exists a measurable map \(P:M\rightarrow GL(d,\mathbb {R})\) satisfying \(A(x)=P(f(x))P(x)^{-1}\) for \(\mu \)-almost every \(x\in M\). Moreover, we prove that whenever the measure \(\mu \) has local product structure the transfer map P is \(\alpha \)-Hölder continuous in sets with arbitrary large measure.


Livšic theorem Non-uniformly hyperbolic systems Matrix-valued cocycles 

Mathematics Subject Classification

Primary 37H15 37A20 Secondary 37D25 



We thank to the anonymous referee for suggestions that helped to improve this presentation. Lucas Backes was partially supported by a CAPES-Brazil postdoctoral fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago. Mauricio Poletti was partially supported by Fondation Louis D-Institut de France (Project coordinated by M. Viana).


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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do Rio Grande do SulPorto AlegreBrazil
  2. 2.LAGA – Université Paris 13VilletaneusFrance

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