Jarek Kwapisz (Montana State University)

Rotation sets and combinatorics of torus diffeomorphisms.

Abstract. The rotation sets of isotopic to the identity diffeomorphisms of the two torus are convex compact subsets of the plane. What sets are realized is not fully understood. Also, even if the rotation set is just a single non-resonant irrational vector, little is known about the map (besides the local KAM type results).

I will present two theorems. The first asserts that the rationally sloped segments that contain no rational points and some irrationally sloped segments cannot be rotation sets (unless they degenerate to points). The second secures existence of renormalization and dynamical partitions of the torus and shows that all maps with a fixed non-resonant irrational vector for their rotation set are combinatorially equivalent.