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

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) 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' 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 that a splitting over R^d is guaranteed even 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.Comment: 26 page

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Producción CientíficaLinear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as well as on a space of p-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.NCN grant Maestro 2013/08/A/ST1/00275MICIIN/FEDER Grant RTI2018-096523-B-100H2020-MSCA-ITN-2014 643073 CRITICS

Conditions for Equality between Lyapunov and Morse Decompositions

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow $\phi$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$. A continuous one given by the finest Morse decomposition and a measurable one furnished by the Multiplicative Ergodic Theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under pertubations leaving unchanged the flow on the base space