A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Gary Froyland,Simon Lloyd,Anthony Quas

doi:10.3934/dcds.2013.33.3835

Abstract

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.

Highlights

Oseledets-type ergodic theorems deal with dynamical systems σ : Ω → Ω where for each ω ∈ Ω there is an operator Lω acting on a linear space X
The first result on the existence of Lyapunov filtrations and Oseledets splittings in the finite dimensional setting is the Multiplicative Ergodic Theorem of Oseledets
If the base is invertible, Lω is invertible a. e. and log+ L±ω 1 dP < +∞, R admits a measurable Oseledets splitting. This situation may be summarized by saying that Oseledets splittings can be found when the base is invertible and the linear actions in the cocycle are invertible with bounded inverses, whereas in the non-invertible linear action cases the theorem only guarantees a Lyapunov filtration

Summary

Introduction

Oseledets-type ergodic theorems deal with dynamical systems σ : Ω → Ω where for each ω ∈ Ω there is an operator (or in the original Oseledets case a matrix) Lω acting on a linear space X. The first result on the existence of Lyapunov filtrations and Oseledets splittings in the finite dimensional setting is the Multiplicative Ergodic Theorem of Oseledets. E. and log+ L±ω 1 dP < +∞, R admits a measurable Oseledets splitting This situation may be summarized by saying that Oseledets splittings can be found when the base is invertible and the linear actions in the cocycle are invertible with bounded inverses, whereas in the non-invertible linear action cases the theorem only guarantees a Lyapunov filtration. Let X be a Banach space and consider a random dynamical system R = (Ω, F , P, σ, X, L) with base transformation σ : Ω → Ω an ergodic homeomorphism, and suppose that the generator L : Ω → L(X, X) is P-continuous and satisfies log+ Lω dP < +∞.

The Grassmannian of a Banach Space

Oseledets splitting

Applications

Application I

Application II