Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 5 May 2020 / Approved: 5 May 2020 / Online: 5 May 2020 (13:21:40 CEST)

How to cite: Azonnahin, A. Dynamics of Cohomological Expanding Mappings II : Third and Fourth Main Results. Preprints 2020, 2020050082 (doi: 10.20944/preprints202005.0082.v1). Azonnahin, A. Dynamics of Cohomological Expanding Mappings II : Third and Fourth Main Results. Preprints 2020, 2020050082 (doi: 10.20944/preprints202005.0082.v1).

## Abstract

Let f : VV be a Cohomological Expanding Mapping1 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. Using pluripotential methods, 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 $\Vc$ . We have also studied the main stochastic properties of $\nu_{h}$ and have shown that $\nu_{h}$ is a smooth equilibrium measure , ergodic, mixing, K-mixing, exponential-mixing. 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. Finally we conjecture that $$\nu_h:=T_l^+ \wedge T_{k-l}^-,$$ $$\dim_\HH(\nu_h)= \Psi h_{\chi}(h) ,$$ $$\dim_\HH( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l},$$ $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1+\log^+{1\over D(x,\Tc)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}$$ and $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1 +\log^+{1\over D(x,E_\gamma)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}.$$

## Supplementary and Associated Material

https://www.preprints.org/manuscript/202004.0521/v1: Dynamics of Cohomological Expanding Mappings I: First and Second Main Results

## Subject Areas

Complex Dynamics; Cohomological Expanding Mapping; Cohomological Degree; Cohomological Entropy; Cohomological Quotient