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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
Article

Abstract

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.

Keywords

Livšic theorem Non-uniformly hyperbolic systems Matrix-valued cocycles 

Mathematics Subject Classification

Primary 37H15 37A20 Secondary 37D25 

Notes

Acknowledgements

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

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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