Entropy of Isometries Semi-groups of Hyperbolic space

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.