Expanding on jc's comment (in particular the reference to Boyland's survey).
One way in which dynamics and low dimensional topology get mixed is via the Nielsen-Thurston classification. The point is that this classification allows one to more or less single out a `simplest' homeomorphism in each isotopy class. Knowing this has allowed people to show that simple topological patterns (usually involving only a finite number of points) can imply very complicated dynamics.
A striking examples of this is given by Llibre and MacKay's theorem for torus homeomorphisms (the existence of 3 non-collinear rotation vectors implies positive entropy).
A nice exposition with several examples is given by Mackay here.
