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 $E$.
Categories: math.DS