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

By Gary Froyland, Simon Lloyd, Anthony Quas. 2010

81 citations recorded by the source corpus.

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.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a7b79e9b9c9c68b02e05cda023967f3f066a7432

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In Control and Systems Engineering

1. C. Gonz'alez-Tokman
2. A. Blumenthal
3. Anthony Quas

In Mathematical Physics

1. A. Blumenthal
2. C. Gonz'alez-Tokman
3. Anthony Quas
4. Gary Froyland

In Statistical and Nonlinear Physics

1. C. Gonz'alez-Tokman
2. A. Blumenthal
3. Anthony Quas
4. Gary Froyland

Citations in the corpus

• A volume-based approach to the multiplicative ergodic theorem on Banach spaces (2015, background): “gmented Banach space. Doan, in his PhD thesis [25], carried out an analogous deduction for strongly measurable cocycles on a separable Banach space, assuming the MET in [16]. Quas, Froyland and Lloyd [9] deduced from the One-Sided MET in [27] that even when the cocycle is not injective, there is sti”
• A semi-invertible operator Oseledets theorem (2011, methodology): “In [19], the proof exploited the structure of the proof given by Thieullen.”
• A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems (2017, unknown): “The quasi-compactness of the twisted transfer operator cocycle (item 2 above) will be based on the works [18,20], which have adapted multiplicative ergodic theory to the setting of cocycles of possibly non-injective operators; the non-injectivity is crucial for the study of endomorphisms Tω.”
• A concise proof of the Multiplicative Ergodic Theorem on Banach spaces (2014, background): “ess condition holds. Separability allows one to show all the necessary measurability conditions in an elementary fashion. The recent semi-invertible version of the MET established in [4], building on [2,3], and from which the invertible version of the MET can be recovered, is also presented here. Th”