Subadditive and Multiplicative Ergodic Theorems

TL;DR

This paper advances ergodic theory by establishing refined subadditive and multiplicative theorems, revealing detailed growth behaviors of random processes and extending classical results to broader contexts.

Contribution

It proves a more delicate subadditive ergodic theorem and generalizes the multiplicative ergodic theorem to include semi-contractions and various applications.

Findings

Growth of random semi-contractions is directed by horofunctions

Generalization of Karlsson-Ledrappier's multiplicative ergodic theorem

Applications to ergodic cocycles, holomorphic maps, and topical operators

Abstract

A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson-Ledrappier, showing that the growth of a random product of semi-contractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.