Ergodicity of cocyles over 2-dimensional rotations

TL;DR

This work investigates the recurrence and ergodicity of cocycles taking values in ${\mathbb R}^d$ over rotations on the torus, with emphasis on badly approximable parameters and discontinuities under Diophantine constraints. It develops a framework based on recurrence times, essential values, and Lebesgue density arguments to construct ergodic cocycles in higher dimensions, including explicit classes $\mathcal F_1$ and $\mathcal F_2$ on ${\mathbb T}^2$. The authors derive recurrence results for broad function classes and provide ergodicity criteria for several cocycle families, along with ergodicity of compact extensions over the triangle $\Delta_0$, under Diophantine and total irrationality assumptions. They also supply technical appendices on Schmidt games and a Lebesgue density theorem variant to support the density-based arguments used throughout. Overall, the paper advances the understanding of when high-dimensional cocycles over toral rotations exhibit recurrence and ergodicity, highlighting the delicate balance between Diophantine properties, discontinuities, and function regularity.

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 $ρ$ , $ρ$ \> 1. The discontinuities of the functions generating the cocycles also satisfy a Diophantine condition. For simplicity of notation we mainly consider the cases $ρ$ = 2, d = 1 and 2.

Figures (1)

• Figure 1: Partition ${\mathcal{P}}_\ell, \ell = 20$ ($\alpha_1= \sqrt 2, \alpha_2=e$)

Theorems & Definitions (35)

• Lemma 1.1
• Lemma 1.2
• Proposition 1.3
• Lemma 2.1
• Proposition 2.2
• Proposition 2.3
• Definition 2.4
• Theorem 2.5
• Proposition 2.6
• Proposition 2.7