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.