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

This paper concerns the compact group extension $$f:\mathbb{T}^2\to \mathbb{T}^2,\quad f (x,s)= (E(x), s+\tau(x)\ \text{mod }1)$$ 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 $\tau$, once we fix $E$.