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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.. (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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Official URL: http://dx.doi.org/10.1007/s10884-011-9213-6

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

Dates:

Date	Event
2011	Published

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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