  Let $f:X \longrightarrow X $ be a Cohomological Hyperbolic Mapping of a
complex compact connected K\"ahler manifold with $ dim_{\mathbb{C}}(X)=k \ge
1$. We want to study the dynamics of such mapping from a probabilistic point of
view, that is, we will try to describe the asymptotic behavior of the orbit $
O_{f} (x) = \{f^{n} (x), n \in \mathbb{N}$ or $\mathbb{Z}\}$ of a generic
point. To do this, using pluripotential methods, we will construct a natural
invariant canonical probability measure of maximum entropy $ \mu_{f} $ such
that $ \lambda_{\max}^{-n}(f^{n})^{\star}\Theta \longrightarrow \mu_{f} $ for
each smooth probability measure $\Theta $ in $X$ with $ \lambda_{k} :=
\limsup_{n\longrightarrow \infty} \{||(f^{n})^{\star}||^{\frac{1}{n}}\} $ the
number of pre-images of a generic point of $X $ by $f $. Then we will study the
main stochastic properties of $ \mu_{f}$ and show, if possible, that $ \mu_{f}$
is a measure of equilibrium, smooth, hyperbolic, ergodic, mixing, $\mathbb{K} $
-mixing, exponential-mixing, moderate and the only measure of maximum entropy,
absolutely continuous with respect to the LEBESGUE measure and to the HAUSDORFF
measure under certain hypotheses. On the other hand, we will introduce the
concept of Perfect and $\mathbb{K} $-Perfect Measure and indeed show that $
\mu_{f}$ is $\mathbb{K} $-Perfect.
