Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 887–908 | Cite as

Hyperbolicity via Evolution Semigroups on \(L^p\)

  • Luis BarreiraEmail author
  • Claudia Valls


For a measurable cocycle over a measurable flow with values on a Banach space, we give a complete characterization of its nonuniform hyperbolicity in terms of an evolution semigroup on \(L^p\). We consider the general cases of a space \(L^p\) determined by a Borel measure that need not be invariant and of an evolution semigroup that need not be strongly continuous, which makes it impossible to use generators. A nontrivial application of our work is the robustness of the hyperbolicity, either uniform or nonuniform, for measurable cocycles over a measurable flow: namely, the robustness of the hyperbolicity for the evolution semigroup yields the robustness of the hyperbolicity for the corresponding cocycle.


Evolution semigroups \(L^p\) spaces Nonuniform hyperbolicity 

Mathematics Subject Classification

Primary 37D99 



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Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

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