A SEMI-INVERTIBLE OSELEDETS THEOREM WITH APPLICATIONS TO TRANSFER OPERATOR COCYCLES

作者：Froyland Gary^*; Lloyd Simon; Quas Anthony

来源：Discrete and Continuous Dynamical Systems, 2013, 33(9): 3835-3860.

DOI：10.3934/dcds.2013.33.3835

• Multiplicative ergodic theorem
• covariant vector
• Lyapunov vector
• equivariant subspace

Oseledets%26apos; 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 R-d. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of R-d into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets%26apos; 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 3 0 (2010),, 729-756.] that a splitting over 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.

• 出版日期2013-9