# The dynamics of contact manifolds

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An interesting question in dynamics is the following: Does any non-singular, volume-preserving flow on a three-manifold $M$ necessarily have a periodic orbit?
The answer is unfortunately no in general. However, in the case where $M$ is a contact manifold and the flow is the standard one known as the Reeb flow, the answer turns out to be yes! We’ll discuss an approach to this problem via the study of holomorphic curves in the four-manifold $\mathbb{R}\times M$.