Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Dynamics of Cohomological Expanding Mappings I: First and Second Main Results

Version 1 : Received: 29 April 2020 / Approved: 30 April 2020 / Online: 30 April 2020 (08:51:44 CEST)

How to cite: Azonnahin, A. Dynamics of Cohomological Expanding Mappings I: First and Second Main Results. Preprints 2020, 2020040521 (doi: 10.20944/preprints202004.0521.v1). Azonnahin, A. Dynamics of Cohomological Expanding Mappings I: First and Second Main Results. Preprints 2020, 2020040521 (doi: 10.20944/preprints202004.0521.v1).

## Abstract

Let $f:V\rightarrow V$ be a Cohomological Expanding Mapping1 of a smooth complex compact homogeneous manifold with dimC(V) = k ≥ 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 $O_{f} (x) = \{f^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. Using pluripotential methods, we construct a natural invariant canonical probability measure of maximum Cohomological Entropy $\mu_{f}$ such that ${\chi_{2l}^{-m}} (f^m)^\ast \Omega \to \mu_f \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega$ on V . Then we study the main stochastic properties of $\mu_{f}$ and show that $\mu_{f}$ is a measure of equilibrium, smooth, ergodic, mixing, K-mixing, exponential-mixing and the unique measure with maximum Cohomological Entropy. We also conjectured that $\mu_f:=T_l^+ \wedge T_{k-l}^-$, $\dim_\H(\mu_f)= \Psi h_{\chi}(f)$ and $\dim_\H( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l}.$

## Keywords

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

## Subject

MATHEMATICS & COMPUTER SCIENCE, Other

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