  We give a generalization to convex co-compact semigroups of a beautiful
theorem of Patterson-Sullivan, telling that the critical exponent (that is the
exponential growth rate) equals the Hausdorff dimension of the limit set (that
is the smallest closed non-empty invariant subset), for a isometries discrete
group of a proper hyperbolic space with compact boundary. To do that, we
introduce a notion of entropy, which generalize the notion of critical exponent
of discrete groups, and we show that it is equal to the upper bound of critical
exponents of Schottky sub-semigroups (which are semigroups having the simplest
dynamic). We obtains several others corollaries, such that the lower
semi-continuity of the entropy, the fact that the critical exponent of a
separate semigroup, that is defined as an upper limit, is in fact a true limit,
and we obtain the existence of "big" Schottky sub-semigroups in discrete groups
of isometries.
