The Multiplicative Ergodic Theorem (MET) is a powerful tool with various applications in different fields of mathematics, including analysis, probability theory, and geometry, and a cornerstone in smooth ergodic theory. Oseledets first proved it for matrix cocycles; since then, the theorem attracted many researchers to present new proofs and formulations with increasing generality.
This talk intends to provide a new version of MET for stationary compositions on a (possibly random) field of (potentially distinct) Banach spaces, depending on the random sample. MET has two versions, and in the first talk, I will concentrate on the one-sided form of this theorem. The primary motivation of this work is to implement a dynamical approach for stochastic delay equations. Analyzing the long-time behavior of this type of equation is a challenging task; since their corresponding solutions often fail to admit the flow property. Our MET, in particular, can be applied to this family of equations to prove the existence of the Lyapunov exponents. In the second part of the talk, which is supposed to be given in the week after, I will talk about the semi-invertible version of MET.
This work is based mainly on the speaker's P.h.D thesis.


[IPM Youth Seminars on Topology and Dynamics]