Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings
Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings
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DOI：
10.1007/978-3-642-13171-4_4
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发表时间：
2008-10
影响因子：
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通讯作者：
T. Dinh;N. Sibony
中科院分区：
文献类型：
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作者：
T. Dinh;N. Sibony
这门入门课程的重点是高维复杂动力学中的多能方法。它们基于多次谐波（p.s.h.）函数的紧致性和正闭合电流理论。这些方法的应用并不局限于我们在这里考虑的动力系统。然而，我们选择证明它们的有效性，并描述两大类映射的理论：射影空间的自同态和类多项式映射。第一部分讨论射影空间的全纯自同态。我们建立了格林电流的第一性质，并给出了平衡测度μ =Tk的几种构造。重点是定量性质和收敛速度。然后我们处理平均分配问题。我们证明了一个完全不变的固有代数集te的存在性，即当∈E时，均匀分布在光纤−n(a)上的概率测度收敛于平衡测度μ，且趋于无穷。对不变子变量的限制也有类似的结果。研究了用任意维变量代替点时的均匀分布问题，讨论了周期点的均匀分布问题。然后，我们建立了μ: k混合的遍历性质，各种可观测值的相关指数衰减，中心极限定理和大偏差定理。我们大量地利用了准p.s.h.差分空间的紧性。功能。特别地，我们证明度量μ是适中的，即⟨μ，eα | φ |⟩≤c，在φ In的有界集合上，对于合适的正常数α，c。最后，我们研究了熵、李亚普诺夫指数和μ的维数。第二部分发展了类多项式映射的理论，即固有全纯映射：U→VwhereU，Vare （kwithVconvex）和U⋐V的开子集。我们引入了这种映射的动态度，构造了最大熵的平衡测度μ。然后，在对动态度的自然假设下，证明了点的等分布性质和测度μ的各种统计性质。假设在地图上的小波动下是稳定的。我们还研究了μ的维数、李亚普诺夫指数及其变化。我们的目标是得到一个独立的文本，只需要一个最小的背景。为了帮助读者，附录给出了射影空间上的p.s.h.函数、正闭合电流和超电位的基础知识。提出了一些练习，并给出了广泛的参考书目。
The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings. The first section deals with holomorphic endomorphisms of the projective space. We establish the first properties and give several constructions for the Green currentsTpand the equilibrium measure μ =Tk. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic setE, totally invariant, i.e., such that whena∉E, the probability measures, equidistributed on the fibersf−n(a), converge towards the equilibrium measure μ, asngoes to infinity. A similar result holds for the restriction offto invariant subvarieties. We survey the equidistribution problem when points are replaced with varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of μ: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the spaceof differences of quasi-p.s.h. functions. In particular, we show that the measure μ is moderate, i.e. ⟨μ,eα | φ |⟩ ≤c, on bounded sets of φ in, for suitable positive constants α,c. Finally, we study the entropy, the Lyapounov exponents and the dimension of μ. The second section develops the theory of polynomial-like maps, i.e. proper holomorphic mapsf:U→VwhereU,Vare open subsets ofℂkwithVconvex andU⋐V. We introduce the dynamical degrees for such maps and construct the equilibrium measure μ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure μ. The assumption is stable under small pertubations on the map. We also study the dimension of μ, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.