On the asymptotic behavior of cocycles over flows

TL;DR

This paper investigates the asymptotic behavior of cocycles over flows, extending classical ergodic theorems by demonstrating convergence modulo finite measure time subsets, thus broadening understanding of cocycle dynamics.

Contribution

It proves that convergence of cocycles over flows occurs not only along subsets of density 1 but also modulo finite measure time subsets, advancing ergodic theory.

Findings

Convergence along subsets of density 1 established

Convergence modulo finite measure time subsets demonstrated

Extends classical results in ergodic theory

Abstract

In 1968, V.I. Oseledets formulated the question of convergence in the Birkhoff theorem and the multiplicative ergodic theorem for measurable cocycles over flows under the condition of integrability for each individual t. A.M. Stepin and the author established (2016) the convergence along subsets of time of density 1. In this note, we show that moreover the convergence is fulfilled modulo time subsets of finite measure.