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Coherent structures and isolated spectrum for Perron–Frobenius cocycles

Published online by Cambridge University Press:  04 September 2009

GARY FROYLAND ,
SIMON LLOYD  and
ANTHONY QUAS
GARY FROYLAND
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney,NSW 2052, Australia (email: g.froyland@unsw.edu.au, s.lloyd@unsw.edu.au)
SIMON LLOYD
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney,NSW 2052, Australia (email: g.froyland@unsw.edu.au, s.lloyd@unsw.edu.au)
ANTHONY QUAS
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria,BC V8W 3R4, Canada (email: aquas@uvic.ca)
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Abstract

We present an analysis of one-dimensional models of dynamical systems that possess ‘coherent structures’: global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalize the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 729 - 756
Copyright
Copyright © Cambridge University Press 2009

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