A volume-based approach to the multiplicative ergodic theorem on Banach spaces

TL;DR

This paper introduces a volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces, extending Ruelle's approach from Hilbert spaces and providing a new interpretation of Lyapunov exponents.

Contribution

It presents a novel volume growth method for proving the theorem in Banach spaces, offering a new perspective on Lyapunov exponents.

Findings

Volume growth interpretation for Lyapunov exponents

Extension of Ruelle's approach to Banach spaces

New proof technique for the Multiplicative Ergodic Theorem

Abstract

A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the Lyapunov exponents of a Banach space cocycle.