A semi-invertible operator Oseledets theorem

TL;DR

This paper introduces a semi-invertible multiplicative ergodic theorem that enables the analysis of transfer operators for randomly composed piecewise expanding interval maps, expanding the applicability of Oseledets splitting.

Contribution

It provides a constructive proof of a semi-invertible ergodic theorem applicable to non-invertible operators like transfer operators in a novel, broad setting.

Findings

Establishes the existence of an Oseledets splitting for certain non-invertible cocycles.

Applies the theorem to transfer operators of randomly composed interval maps.

Demonstrates the theorem's utility in dynamical systems analysis.

Abstract

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum.