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On statistical attractors and the convergence of time averages

Published online by Cambridge University Press:  12 January 2011

ÖZKAN KARABACAK  and
PETER ASHWIN
ÖZKAN KARABACAK
Affiliation:
Mathematics Research Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF. e-mail: O.karabacak@exeter.ac.uk and P.ashwin@exeter.ac.uk
PETER ASHWIN
Affiliation:
Mathematics Research Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF. e-mail: O.karabacak@exeter.ac.uk and P.ashwin@exeter.ac.uk
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Abstract

There are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 353 - 365
Copyright
Copyright © Cambridge Philosophical Society 2011

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