Dynamics of Cohomological Expanding Mappings II :Third and Fourth Main Results

Let $h:\Vc \longrightarrow \Vc $ be a Cohomological Expanding Mapping of a smooth complex compact homogeneous manifold with $ dim_{\mathbb{C}}(\Vc)=k \ge 1$ and Kodaira Dimension $\leq 0$. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit $ O_{h} (x) = \{h^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. We have constructed in our previous paper \cite{Armand4} a natural invariant canonical probability measure of maximal Cohomological Entropy $ \nu_{h} $ such that ${\chi_{2l}^{-m}} (h^m)^\ast \Omega \to \nu_h \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega $ in $\Vc$ . In this paper we are interested on equidistribution problems and we show in particular that $ \nu_{h}$ reflects a property of equidistribution of periodic points by setting out the Third and Fourth Main Results in our study.