Invariant Measures for Dissipative Systems and Generalised Banach Limits
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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.
KeywordsInvariant measure Generalised Banach limit Krylov–Bogoliubov procedure Global attractor
We would like to thank the referees for their helpful comments; one pointed out that Theorem 11 holds in any metric space, which led to a strengthening of many of our results. GŁ was supported by Polish Government Grant MNiSW N N201 547638. JR was supported by Spanish Ministerio de Ciencia e Innovación, Project MTM2008-00088, and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía), Proyecto de Excelencia P07-FQM-02468. JCR was supported by an EPSRC Leadership Fellowship, grant EP/G007470/1.
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