  We propose a definition for the length of closed geodesics in a globally
hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that
the number of closed geodesics of length less than $R$ grows exponentially fast
with $R$ and the exponential growth rate is related to the critical exponent
associated to the two hyperbolic surfaces coming from Mess parametrization. We
get an equivalent of three results for quasi-Fuchsian manifolds in the GHMC
setting : R. Bowen's rigidity theorem of critical exponent, A. Sanders'
isolation theorem and C. McMullen's examples lightening the behaviour of this
exponent when the surfaces range over Teichm\"uller space.
