Seminar

Department of Mathematics

Monday, March 24th, 2008 at 3PM

BA6183, 40 St. George Street

DYNAMICAL SYSTEMS SEMINAR -- Some properties of homeomorphisms of the two-torus.

Andres Koropecki

IM-UFF Rio de Janiero

We deal with homeomorphisms of the two-dimensional torus in the isotopy class of the identity. We will show results about the existence of periodic points, and on the other hand we will also talk about the topological structure of minimal systems.

Our first result shows that if there are no free curves (i.e. essential simple closed curves disjoint from its image by the map) then either there is a fixed point, or the topological entropy is positive (and there are infinitely many periodic points). This is done analyzing the structure of the rotation set (a generalization of the rotation number for circle homeomorphisms) and its relationship with the existence of free curves.

Later we analyze the dynamical consequences in case that there are "always free" curves, and we discuss if the existence of this type of curves is a typical phenomena among systems without periodic points. To do that we discuss generic properties of minimal systems.

Most of this talk is a joint work with Alejandro Kocsard.