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. Preprints2020, 2020040521. https://doi.org/10.20944/preprints202004.0521.v1
Azonnahin, A. Dynamics of Cohomological Expanding Mappings I: First and Second Main Results. Preprints 2020, 2020040521. https://doi.org/10.20944/preprints202004.0521.v1
Azonnahin, A. Dynamics of Cohomological Expanding Mappings I: First and Second Main Results. Preprints2020, 2020040521. https://doi.org/10.20944/preprints202004.0521.v1
APA Style
Azonnahin, A. (2020). Dynamics of Cohomological Expanding Mappings I: First and Second Main Results. Preprints. https://doi.org/10.20944/preprints202004.0521.v1
Chicago/Turabian Style
Azonnahin, A. 2020 "Dynamics of Cohomological Expanding Mappings I: First and Second Main Results" Preprints. https://doi.org/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 $\dim_\H( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l}.$
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
