Article

Journal of Dynamics and Differential Equations

, Volume 23, Issue 2, pp 225-250

Open Access This content is freely available online to anyone, anywhere at any time.

Invariant Measures for Dissipative Systems and Generalised Banach Limits

  • Grzegorz ŁukaszewiczAffiliated withMathematics Department, University of Warsaw
  • , José RealAffiliated withDepartamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas
  • , James C. RobinsonAffiliated withMathematics Institute, University of Warwick Email author 

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