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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 expandingmap $E:\mathbb{S}^1\to \mathbb{S}^1$. The dynamics of $f$ and its stochasticperturbations have previously been studied under the so-called partialcaptivity condition. Here we prove a supplementary result that shows thatpartial captivity is a $\mathscr{C}^r$ generic condition on $\tau$, once we fix$E$.

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