2010/01/28 by Gary Froyland, S. P. Lloyd, Froyland, Gary +4 · 1 citation
Engineering · Mathematics · #37A30 #37H15 (Primary) #37L55 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1001.5313
openalex publication_date 2010/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with\nthe exponential growth rates of vectors under the action of a linear cocycle on\nRd. When the linear actions are invertible, the MET guarantees an\nalmost-everywhere pointwise splitting of Rd into subspaces of distinct\nexponential growth rates (called Lyapunov exponents). When the linear actions\nare non-invertible, Oseledets' MET only yields the existence of a filtration of\nsubspaces, the elements of which contain all vectors that grow no faster than\nexponential rates given by the Lyapunov exponents. The authors recently\ndemonstrated that a splitting over Rd is guaranteed even without the\ninvertibility assumption on the linear actions. Motivated by applications of\nthe MET to cocycles of (non-invertible) transfer operators arising from random\ndynamical systems, we demonstrate the existence of an Oseledets splitting for\ncocycles of quasi-compact non-invertible linear operators on Banach spaces.\n