The partial captivity condition for U(1) extensions of expanding maps on the circle

This paper concerns the compact group extension $f:\mathbb{T}^2\to \mathbb{T}^2,\quad f (x,s)= (E(x), s+τ(x)\ \text{mod }1)$ of an expanding map $E:\mathbb{S}^1\to \mathbb{S}^1$. The dynamics of $f$ and its stochastic perturbations have previously been studied under the so-called partial captivity condition. Here we prove a supplementary result that shows that partial captivity is a $\mathscr{C}^r$ generic condition on $τ$, once we fix $E$.

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