Coherent structures and isolated spectrum for Perron-Frobenius cocycles

TL;DR

This paper analyzes one-dimensional dynamical systems with coherent structures, proving finite isolated Lyapunov spectrum points for certain cocycles and developing a new formalism for identifying coherent structures in non-autonomous systems.

Contribution

It introduces a novel analysis of Perron-Frobenius cocycles with piecewise affine generators, establishing finite isolated spectrum points and constructing invariant Oseledets subspaces.

Findings

Finite number of isolated Lyapunov spectrum points.

Construction of invariant Oseledets subspaces.

Identification of coherent structures in non-autonomous systems.

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 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.