On $SL(2,\mathbb{R})$-cocycles over irrational rotations with secondary collisions

TL;DR

This paper investigates the hyperbolic behavior of certain $SL(2, eal)$-cocycles over irrational rotations, showing that secondary collisions can restore hyperbolicity even when primary collisions weaken it.

Contribution

It introduces a method to demonstrate how secondary collisions can compensate for primary collision effects, ensuring hyperbolicity in specific skew product systems.

Findings

Secondary collisions can restore hyperbolicity.

Hyperbolicity depends on the derivative of the rotation angle function.

Conditions on the zeroes of $ ext{cos} \, ext{varphi}$ are crucial.

Abstract

We consider a skew product $F_{A} = (\sigma_{\omega}, A)$ over irrational rotation $\sigma_{\omega}(x) = x + \omega$ of a circle $\mathbb{T}^{1}$. It is supposed that the transformation $A: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ being a $C^{1}$-map has the form $A(x) = R(\varphi(x)) Z(\lambda(x))$, where $R(\varphi)$ is a rotation in $\mathbb{R}^{2}$ over the angle $\varphi$ and $Z(\lambda)= diag\{\lambda, \lambda^{-1}\}$ is a diagonal matrix. Assuming that $\lambda(x) \ge \lambda_{0} > 1$ with a sufficiently large constant $\lambda_{0}$ and the function $\varphi$ be such that $\cos \varphi(x)$ possesses only simple zeroes, we study hyperbolic properties of the cocycle generated by $F_{A}$. We apply the critical set method to show that, under some additional requirements on the derivative of the function $\varphi$, the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by $F_{A}$ becomes hyperbolic in contrary to the case when secondary collisions can be partially eliminated.