Torus diffeomorphisms with parabolic and non-proper actions on the fine curve graph and their generalized rotation sets

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