$\omega$-recurrence in cocycles

TL;DR

This paper investigates $ ext{omega}$-recurrence in cocycles over irrational rotations, linking it to ergodic sums and Lyapunov exponents, and identifies conditions under which cocycles exhibit positive Lyapunov exponents.

Contribution

It provides a detailed analysis of $ ext{omega}$-recurrence in ergodic cocycles, showing generic $1/n$-recurrence and constructing examples with positive Lyapunov exponents for certain decay rates.

Findings

Generic cocycles are $1/n$-recurrent.

Existence of uncountably many cocycles with positive Lyapunov exponents.

Conditions under which cocycles are not $ ext{omega}$-recurrent.

Abstract

After relating the notion of $\omega$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $\omega(n) <n^{-\epsilon}$, where $\epsilon>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $\omega$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over interval exchange transformations of periodic type.