On Cohomological Stochastic Dynamics of Higher Dimensions
Résumé
Here is one of the best and the more recent books on Complex Dynamics in Higer Dimension. Here, we develop a calculus on positive closed currents of arbitrary bidegree on compact K{\"a}hler manifolds after recalling a notion of super-potential associated to such currents by using deformations in the space of currents. We define in particular the intersection of such currents and the pull-back operator. We apply the theory of super-potentials to construct Green currents and to study distribution problems . We obtain several results on the stochastic dynamics such as the regularity and the uniqueness of the Green currents, the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures.We obtain many stochastic properties for the equilibrium measures. We prove the exponential decay of correlations, the central limit theorem and the large deviations theorem. One of the main tools is the use of structural discs in the space of positive closed currents which gives a "geometry" on that space. As an application, for example, when $f$ is a dominating meromorphic self-map of large topological degree on a compact K\"ahler manifold , we give the recent new construction of the equilibrium measure $\mu_{f}$ of $f$ , prove that $\mu_{f}$ is exponentially mixing , show that repelling periodic points are equidistributed with respect to $\mu_{f}$ and describe the exceptional set of points whose backward orbits are not equidistributed with respect to $\mu_{f}$ .Finally, we prove exponential estimates for plurisubharmonic functions with respect to Monge-Amp{\`e}re measures and discuss some open value distribution problems .
Mots clés
- periodic point
- Complex dynamics
- large deviations theorem.
- central limit theorem
- exponential mixing
- Monge-Amp{\`e}re measure
- tangent current
- exceptional set
- value distribution
- dynamical degrees
- meromorphic self-map
- $p$-stable maps
- regular automorphism
- pull-back operator
- intersection theory
- disc of currents
- equilibrium measure
- moderate measure
- variety of currents
- super-potential
- Green current
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