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

TL;DR

This paper demonstrates that generic torus diffeomorphisms exhibit parabolic, non-proper actions on the fine curve graph and possess generalized rotation sets of various symmetric convex types, revealing complex dynamical behaviors.

Contribution

It establishes the generic dynamical properties of torus diffeomorphisms in the Anosov-Katok class regarding their actions on the fine curve graph and their rotation sets.

Findings

Generic elements act parabolically on the fine curve graph.

Generic elements have diverse generalized rotation sets.

The results connect dynamical actions with geometric rotation set types.

Abstract

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.