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The Lyapunov exponents of generic skew-product compact semiflows

  • Mário BessaEmail author
  • Glória Ferreira Carvalho
Article
  • 6 Downloads

Abstract

Let \({\mathscr {F}}_K\) denote the set of infinite-dimensional cocycles over a \(\mu \)-ergodic flow \(\varphi ^t:M\rightarrow M\) and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset \({\mathscr {R}}\) of \({\mathscr {F}}_K\) such that for \(\Phi \in {\mathscr {R}}\) and for \(\mu \)-almost every \(x\in M\), either:
  1. (i)

    the limit operator \(\underset{t\rightarrow \infty }{\lim }((\Phi ^t(x))^*\Phi ^t(x))^{\frac{1}{2t}}\) is the null operator or else

     
  2. (ii)

    the Oseledets–Ruelle splitting of \(\Phi \) along the \(\varphi ^t\)-orbit of x has a dominated splitting.

     

Keywords

Oseledets–Ruelle theorem Lyapunov exponents Compact semiflows 

Mathematics Subject Classification

Primary 37C40 37D25 37D30 Secondary 47D06 34G10 

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Notes

Acknowledgements

The authors would like to thank the anonymous referee for the careful reading of the manuscript and for giving helpful comments and suggestions. MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia,’ through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. GC was supported by FCT, PhD Scholarship number SFRH/BD/75746/2011. The authors would like to thank CMUP for providing the necessary conditions in which this work was also developed and also António Bento for useful suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universidade da Beira InteriorCovilhãPortugal
  2. 2.FCUPUniversidade do PortoPortoPortugal

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