Journal of Evolution Equations

, Volume 19, Issue 2, pp 387–409 | Cite as

The Lyapunov exponents of generic skew-product compact semiflows

  • Mário BessaEmail author
  • Glória Ferreira Carvalho


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.



Oseledets–Ruelle theorem Lyapunov exponents Compact semiflows 

Mathematics Subject Classification

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


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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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