Coherent structures and isolated spectrum for Perron–Frobenius cocycles

By Gary Froyland, Simon Lloyd, Anthony Quas. Ergodic Theory and Dynamical Systems, 2008

102 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0d276e9e3720a225f3ee659204e404afc905a4aa

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Citations in the corpus

A semi-invertible Oseledets Theorem with applications to transfer operator cocycles (2010, unknown): “The authors discuss and provide examples of such spectral values and Oseledets subspaces in [10].”
Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles (2013, unknown): “1 is not stated in [24].”
A concise proof of the Multiplicative Ergodic Theorem on Banach spaces (2014, background): “The recent semi-invertible version of the MET established in [4], building on [2, 3], and from which the invertible version of the MET can be recovered, is also presented here.”
A semi-invertible operator Oseledets theorem (2011, methodology): “In [18], this was done in a relatively natural way (by pushing forward the orthogonal complement of Vi (σ−nω) Vi+1(σω) under A(σ−1ω) · · · A(σ−nω) and taking a limit as n tends to infinity).”
Dimension of invariant measures for affine iterated function systems (2019, background): “12 is not stated in [32].”
An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems (2012, background): “The finite-time constructions considered here are to be contrasted with recent timeasymptotic transfer operator constructions [17, 18, 24], where equivariant Oseledets subspaces play the role of singular vectors in the present paper.”