Colloquium: “Topological Properties of Orbit Sets for Groups of Homeomorphisms”

Marshall University Math Colloquium
February 22, 2005

Norah Esty
University of California – Berkeley

In this talk, I will introduce some dynamical systems by looking at some examples of the way a homeomorphism can iterate a point on the circle, from fixing points and having periodic orbits to creating dense orbits. I will go over the complete classification of orbit types given by the Poincare Classification Theorem, including the existence of homeomorphisms with an invariant Cantor set. Then I will discuss the analogy between the iterate set for a single homeo (corresponding to an action of the group Z) and the orbit set of more general groups of homeomorphisms. Sacksteder’s Theorem gives three possibilities for a particular group G. Sacksteder’s first case is the existence of a finite orbit, however, it does not give nay information about the orbit type of the remaining points. My work expands this case to examine what happens to all others point on the circle when the group has a common periodic set.