Geometry of Rotation Sets (22rit506)


(The City College of New York)

(Jagiellonian University in Kraków)


The Banff International Research Station will host the "Geometry of Rotation Sets" workshop in Banff from September 25, 2022 to October 2, 2022.

We study the dynamics of homeomorphisms on a two-dimensional torus. The asymptotic motion of orbits of such homeomorphisms can be described using vectors on a plane: the magnitude of the vector gives the speed of motion, and its direction gives a homology class which best approximates the motion. The set of all vectors realized by the orbits of the homeomorphism is called a rotation set and provides a blueprint of the overall dynamics of the system. A question that has remained unanswered for over twenty years is how to characterize all the rotation sets of torus homeomorphisms.

A classical theorem by Misiurewicz and Ziemian from 1989 states that such sets are compact and convex. To this day the only examples of rotation sets with nonempty interior are polygons with vertices at rational coordinates and sets that have infinitely many rational polygonal vertices accumulating on one or two points. The goal of this project is to show that polygons with vertices at irrational points can be realized as a rotation set of a torus homeomorphism. This result would be a significant step towards a resolution of the 30 year old conjecture that any convex compact set on the plane with non-empty interior is a rotation set of a torus homeomorphism.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. BIRS is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).