Lyapunov exponents for families of rotated linear cocycles

TL;DR

This paper investigates how the upper Lyapunov exponent varies analytically with rotation parameters in families of linear cocycles, revealing conditions for dominated splitting and the existence of parameters lacking it.

Contribution

It establishes the real analyticity and strict concavity of the Lyapunov exponent function for rotated cocycles with dominated splitting, and shows the non-emptiness of parameters without dominated splitting.

Findings

Lyapunov exponent function is real analytic and strictly concave on an open set of parameters.

Existence of parameters where the cocycle does not have dominated splitting.

Dominated splitting persists under small rotations in a non-empty parameter set.

Abstract

In this work, we are interested in the study of the upper Lyapunov exponent $\lambda^+(\theta)$ associated to the periodic family of cocycles defined by $$A_\theta(x):=A(x)R_\theta,\qquad x\in X,$$ where $A\::\: X\to \mathbb{GL}^+(2,\mathbb{R})$ is a linear cocycle orientation--preser\-ving and $R_\theta$ is a rotation of angle $\theta\in\mathbb{R}$. We show that if the cocycle $A$ has dominated splitting, then there exists a non empty open set $\mathcal{U}$ of parameters $\theta$ such that the cocycle $A_\theta$ has dominated splitting and the function $\mathcal{U}\ni\theta\mapsto\lambda^+(\theta)$ is real analytic and strictly concave. As a consequence, we obtain that the set of parameters $\theta$ where the cocycle $A_\theta$ has not dominated splitting is non empty.