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 almostcyclic sets to non-autonomous dynamical systems and provide a new ensemblebased formalism for coherent structures in one-dimensional non-autonomous dynamics.
Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.
Given a factor code π from a one-dimensional shift of finite type X onto an irreducible sofic shift Y , if π is finite-to-one there is an invariant called the degree of π which is defined the number of preimages of a typical point in Y . We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure ν on Y , we find an invariant upper bound on the number of ergodic measures on X which project to ν and have maximal entropy among all measures in the fibre π −1 {ν}. We show that this bound and the class degree of the code agree when ν is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy. * We thank the referee for detailed and helpful comments.
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