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We present an analysis of one-dimensional models of dynamical systems thatpossess 'coherent structures'; global structures that disperse more slowly thanlocal trajectory separation. We study cocycles generated by expanding intervalmaps and the rates of decay for functions of bounded variation under the actionof the associated Perron-Frobenius cocycles.  We prove that when the generators are piecewise affine and share a commonMarkov partition, the Lyapunov spectrum of the Perron-Frobenius cocycle has atmost finitely many isolated points. Moreover, we develop a strengthened versionof the Multiplicative Ergodic Theorem for non-invertible matrices and constructan invariant splitting into Oseledets subspaces.  We detail examples of cocycles of expanding maps with isolated Lyapunovspectrum and calculate the Oseledets subspaces, which lead to an identificationof the underlying coherent structures.  Our constructions generalise the notions of almost-invariant andalmost-cyclic sets to non-autonomous dynamical systems and provide a newensemble-based formalism for coherent structures in one-dimensionalnon-autonomous dynamics.

Coherent structures and isolated spectrum for Perron-Frobenius cocycles