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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!

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    $\begingroup$ I just edited the parts of your question that seemed mostly argumentative, but you may want to rewrite the question in order to make it more concrete. A list of important and, indeed, fundamental applications of dynamics to other areas of mathematics from number theory to topology would be huge. $\endgroup$
    – alvarezpaiva
    Commented May 30, 2014 at 10:48
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    $\begingroup$ The title of the question does not seem to be very well aligned with its content: I drifted in here expecting to read about applications to Ramsey theory, spectral theory, Diophantine approximation, analysis of algorithms... $\endgroup$
    – Ian Morris
    Commented May 30, 2014 at 11:39
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    $\begingroup$ Ergodic theory has applications basically everywhere - density of primes in number theory, rigidity theory for lattices in algebraic groups, all over the place in Riemannian geometry (geodesic flow on nonpositively curved manifolds), probability theory... $\endgroup$
    – Paul Siegel
    Commented May 30, 2014 at 12:03
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    $\begingroup$ I have several truly remarkable examples but they all concern holomorphic dynamics (not Hamiltonian or Reeb). If this is also allowed I will write about them. $\endgroup$
    – Alexandre Eremenko
    Commented May 30, 2014 at 12:17
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    $\begingroup$ @AlexandreEremenko: "I have discovered a truly marvellous example of this, which the scope of this question is too narrow to contain..." $\endgroup$
    – Ian Morris
    Commented May 30, 2014 at 12:34

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There is a lot of influence in negative curvature through the geodesic flow. Here are two examples from the top of my head:

1) Farrell-Jones "asymptotic/focal transfer tool" (= geodesic flow on the unit tangent bundle of M), which is needed for one of the major steps (construction of controlled h-cobordism) their celebrated topological rigidity theorem: If M is a non-positively curved closed manifold of dimension >4 and f:M->N is a homotopy equivalence then f is homotopic to a homeomorphism.

2) For Otal/Croke result on marked length spectrum rigidity for negatively curved surfaces the geodesic flow is instrumental. In particular, coincidence of marked length spectra immediately imply existence of a true conjugacy for the geodesic flows via Livshits theorem.

Adding a recent one: Kahn-Marcovic proof of abundance of surface subgroups in hyperbolic 3-manifolds uses dynamics of 2-frame flows.

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Here is an application with which I have first hand experience.

In the sixties J.J. Schaeffer conjectured that the girth of a normed space---the infimum of the lengths of all continuous curves on its unit sphere that join a pair of antipodal points---equals the girth of its dual space.

The normed space is not necessarily complete nor finite-dimensional. Schaeffer proved the conjecture in dimension two (a fun thing to do even in the case of $\ell_p$) and showed that it sufficed to prove the conjecture in finite dimensions.

In this paper the conjecture is settled using some very simple Hamiltonian dynamics. The proof is in slick modern language, but a little tinkering gives a proof that would have been immediately comprehensible to Birkhoff and Poincaré. An added bonus: the same proof gives Cauchy's integral-geometric formula and its generalization to all finite-dimensional normed spaces.

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