The Lyapunov exponents of generic skew-product compact semiflows

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.

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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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Bessa, M., Carvalho, G.F. The Lyapunov exponents of generic skew-product compact semiflows. J. Evol. Equ. 19, 387–409 (2019). https://doi.org/10.1007/s00028-019-00479-8

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Keywords

  • Oseledets–Ruelle theorem
  • Lyapunov exponents
  • Compact semiflows

Mathematics Subject Classification

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