**I'm looking for examples that highlight how dynamical systems (particularly, Hamiltonian and Reeb dynamics) can be used to shed light in other areas of mathematics.** This could potentially include proposed connections in the form of conjectures, etc.

As an example of the sort of ideas I'm looking for: Hofer-Zehnder capacities are defined in terms of the dynamics of certain Hamiltonian functions on a symplectic manifold; in turn capacities can be used to prove several nice results including the Gromov non-squeezing theorem which is an important result in symplectic topology. Capacities are the only example I know of dynamics being exploited in (symplectic) topology. As I pointed out, I am no expert in this field so I'm sure more examples exist.

A more concrete question (focused on topology) could be the following:

**What do (Hamiltonian, Reeb) dynamics tell us about the (symplectic, contact) topology of a (symplectic, contact) manifold?**

EDIT: As was pointed out in the comments this might be a bit of a misleading question. Examples coming from different fields are most welcome, although I am more interested in applications to topology it would also be nice to hear about other areas. However, let me just narrow the category to differentiable dynamical systems.

Thanks!