A note on the marginal instability rates of two-dimensional linear cocycles

TL;DR

This paper revisits a theorem on the growth rates of 2D linear cocycles, providing an alternative proof and showing that the theorem's conclusions do not extend to Lipschitz continuous cocycles.

Contribution

It offers an alternative proof of a known theorem and demonstrates the limitations of its conclusions for Lipschitz continuous cocycles.

Findings

The growth of locally constant GL(2,R)-cocycles is either bounded or linear if not exponential.

The theorem's conclusions do not hold for Lipschitz continuous cocycles.

Provides insights into the behavior of cocycles over full shifts.

Abstract

A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant GL(2,R)-cocycle on the full shift is not exponential then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.