Invariant Measures for Dissipative Systems and Generalised Banach Limits

  • Grzegorz Łukaszewicz
  • José Real
  • James C. Robinson
Open Access
Article

Abstract

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.

Keywords

Invariant measure Generalised Banach limit Krylov–Bogoliubov procedure Global attractor 

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Grzegorz Łukaszewicz
    • 1
  • José Real
    • 2
  • James C. Robinson
    • 3
  1. 1.Mathematics DepartmentUniversity of WarsawWarsawPoland
  2. 2.Departamento de Ecuaciones Diferenciales y Análisis NuméricoFacultad de MatemáticasSevilleSpain
  3. 3.Mathematics InstituteUniversity of WarwickCoventryUK

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