  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 generalise 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.
