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, Ō∞(T²), acts parabolically and non-properly on the fine curve graph C^(T²). Additionally, we show that a generic element of Ō∞(T²) admits generalized rotation sets of any point-symmetric compact convex homothety type in the plane.