  On one side, from the properties of Floer cohomology, invariant associated to
a symplectic manifold, we define and study a notion of symplectic hyperbolicity
and a symplectic capacity measuring it. On the other side, the usual notions of
complex hyperbolicity can be straightforwardly generalized to the case of
almost-complex manifolds by using pseudo-holomorphic curves. That's why we
study the links between these two notions of hyperbolicities when a manifold is
provided with some compatible symplectic and almost-complex structures. We
mainly explain how the non-symplectic hyperbolicity implies the existence of
pseudo-holomorphic curves, and so the non-complex hyperbolicity. Thanks to this
analysis, we could both better understand the Floer cohomology and get new
results on almost-complex hyperbolicity. We notably prove results of stability
for non-complex hyperbolicity under deformation of the almost-complex structure
among the set of the almost-complex structures compatible with a fixed
non-hyperbolic symplectic structure, thus generalizing Bangert theorem that
gave this same result in the special case of the standard torus.
