An exotic flow on a compact surface
In 1988 Anosov  published the construction of an example of a flow (continuous real action) on a cylinder or annulus with a phase portrait strikingly different from our normal experience. It contains orbits whose $οmega$-limit sets contain a non-periodic orbit along with a simple closed curve of fixed points, but these orbits do not wrap down on this simple closed curve in the usual way. In this paper we modify some of Anosov's methods to construct a flow on a surface of genus $2$ with equally striking behavior that does not occur on a surface of genus $1$ or a cylinder. Moreover, our construction is relatively simple and can easily be modified to produce a variety of examples exhibiting similar types of behavior. The key idea that we use from Anosov's paper can be described in the following way. A flow on a cylinder can always be slowed down near one of the boundary circles so that it becomes fixed. If you slow a flow down very rapidly as you approach a bounding circle, then you can also spin the orbits further and further around the axis of the cylinder as you approach the boundary without destroying the flow. In particular, the boundary circle remains fixed, but orbits that approach even a single point on it now spiral toward it.