Din\^amica de Aplica\c{c}\~oes Cohomologicamente Hiperb\'olicas

TL;DR

This paper studies the dynamics of cohomologically hyperbolic mappings on complex Kähler manifolds, constructing a unique invariant measure with maximal entropy and analyzing its stochastic and ergodic properties.

Contribution

It introduces a natural invariant measure for such mappings and investigates its properties, including hyperbolicity, ergodicity, mixing, and absolute continuity, along with new concepts of Perfect and -Perfect measures.

Findings

Constructed a canonical measure of maximum entropy f.

Proved the measure is ergodic, mixing, and hyperbolic.

Established conditions for absolute continuity with Lebesgue and Hausdorff measures.

Abstract

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.