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Invariant measures for dissipative systems and generalised Banach limits

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Łukaszewicz, Grzegorz, Real, José and Robinson, James C. (James Cooper), 1969-. (2011) Invariant measures for dissipative systems and generalised Banach limits. Journal of Dynamics and Differential Equations, Volume 23 (Number 2). pp. 225-250. ISSN 1040-7294

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Invariant measures, Banach spaces, Dynamics, Attractors (Mathematics)
Journal or Publication Title: Journal of Dynamics and Differential Equations
Publisher: Springer
ISSN: 1040-7294
Official Date: 2011
Volume: Volume 23
Number: Number 2
Page Range: pp. 225-250
Identification Number: 10.1007/s10884-011-9213-6
Status: Peer Reviewed
Publication Status: Published
Funder: Poland , Spain. Ministerio de Ciencia e Innovación (MICINN), Andalusia (Spain). Consejería de Innovación, Ciencia y Empresa , Engineering and Physical Sciences Research Council (EPSRC)
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URI: http://wrap.warwick.ac.uk/id/eprint/41549

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