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$ω$-recurrence in cocycles

After relating the notion of $ω$-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 $ω(n) <n^{-ε}$, where $ε>1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $ω$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over interval exchange transformations of periodic type.