Rotation theory and the fine curve graph on the torus: non-proper parabolic examples
The fine curve graph of a surface was introduced by Bowden, Hensel, and Webb in 2022. This hyperbolic metric graph is used to study the homeomorphism group of the surface through its action on the graph.
Since the introduction of this object, many researchers have worked to establish connections between the dynamics of surface homeomorphisms and the dynamics of their action by isometries on the fine curve graph. These endeavors have led to the discovery of a connection between this subject and the rich area of rotation theory on surfaces. Rotation theory was first developed in the late 1980s by Misiurewicz and Ziemian and remains an active field of research to this day.
In this talk, we will introduce the fine curve graph and rotation theory on the two-dimensional torus. We will then see how, by using the famous approximation by conjugation method of Anosov and Katok, we can construct examples of torus diffeomorphisms that act parabolically and non-properly on the fine curve graph, while admitting a rich family of generalized rotation sets.
