Mathematical Proceedings of the Cambridge Philosophical Society

Table of Contents - March 2011 - Volume 150, Issue 02  

Research Article

On statistical attractors and the convergence of time averages

ÖZKAN KARABACAKa1 and PETER ASHWINa1

a1 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

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.

(Received March 30 2010)

(Revised July 04 2010)

(Online publication January 12 2011)