Ergodicity of cocyles over 2-dimensional rotations

TL;DR

This paper investigates the recurrence and ergodic properties of cocycles over 2-dimensional rotations with Diophantine conditions, focusing on cases where the rotation number is badly approximable.

Contribution

It provides new results on the ergodicity of cocycles over rotations with Diophantine conditions, especially for specific dimensions and rotation parameters.

Findings

Establishes conditions for recurrence and ergodicity of cocycles

Analyzes cases with specific rotation parameters and dimensions

Provides theoretical insights into cocycle behavior over rotations

Abstract

We study recurrence and ergodicity of cocycles with values in R d , d $\ge$ 1, over rotations by badly approximable irrational numbers on T $\rho$ , $\rho$ \> 1. The discontinuities of the functions generating the cocycles also satisfy a Diophantine condition. For simplicity of notation we mainly consider the cases $\rho$ = 2, d = 1 and 2.