  We prove that a generic element of the Anosov-Katok class of the torus,
$\overline{\mathcal{O}}^{\infty}(\mathbb{T}^2)$, acts parabolically and
non-properly on the fine curve graph $C^{\dagger}(\mathbb{T}^2)$. Additionally,
we show that a generic element of
$\overline{\mathcal{O}}^{\infty}(\mathbb{T}^2)$ admits generalized rotation
sets of any point-symmetric compact convex homothety type in the plane.
