Thomas Ohlson Timoudas: Dynamical rigidity of circle homeomorphisms (and diffeomorphisms)

Much is known about the dynamical properties of circle homeomorphisms, since the early works of Poincaré and Denjoy. Namely, for an orientation-preserving homeomorphism f: S -> S, from the circle to itself, we can say a lot about the possible structure of its orbits, that is the sets {x, f(x), f(f(x)), …}.

An example of such a map f is the circle rotation R: S -> S, given by R(x) = x + a (mod 1), where a is some real number, and the circle is viewed as [0,1] with its edges identified. If a = p/q is rational, its orbits are all periodic with period q. If a is irrational, then every orbit is dense and uniformly distributed on the circle. Can the orbit structure of a more general such map f:S -> S be much more complicated than this?

If we assume that the map f has a little more regularity (differentiable with some extra conditions), the possible dynamical structures will be surprisingly (!) restricted. I will sketch some of the main ideas behind some classical and new results in this direction.

After this talk, you will be able to answer almost every question (with respect to some suitably ad hoc defined measure) about circle homeomorphisms.