In this talk, we will describe properties of a homeomorphism with no
periodic points on T^2. One topological way is to study its minimal sets,
while another way is to understand its rotation sets.
We will define rotation sets and discuss two different cases. The first
case is when the rotation set is a singleton (which we call a
pseudo-rotation). A natural question is to ask if a totally irraional
pseudo-rotation f admits a unique minimal set. While the answer to this
question is in general no, we obtain results for proper conditions such
that the system has a unique minimal set.
Another case is when the rotation set is a non-trivial line segment. We
will state Franks-Misiurewicz conjecture and introduce recent progress.
With a non-trivial rotation segment, it is also possible that the rotation
vector is almost everywhere not well-defined.
These are part of joint works in progresses with Artur Avila, Martin Leguil
and Disheng Xu.