全纯映射族的动力学性质
Dynamical properties of families of holomorphic mapping
摘要：
本文第一部分研究了紧致度量空间$M$上纤维化的$\mbb C^2$的Hénon映射的斜积的动力学性质。这个问题归结为理解Hénon映射的伪随机序列组成的动力学行为。与单个Hénon映射的迭代动力学类似，可以构造满足适当不变性性质的纤维格林函数以及相应的稳定和不稳定电流。此外，它示出了一个合适的电流由斜Hénon映射的连续拉回收敛到纤维稳定电流的倍数。论文的第二部分将上述结果推广到完全随机的Hénon映射序列。此外，对于这个随机Hénon映射系统，我们引入了平均格林函数和平均格林流的概念，它们具有经典格林函数和格林流的许多典型特征.第三部分是关于一类特殊斜映射的全局动力学的一些结果。为了证明这些结果，我们广泛地利用了Hénon映射的伪随机序列的动力学行为的知识。我们证明了对于一类不变测度，整体斜映射是强混合的，并且给出了斜积的拓扑熵的一个下界。最后，我们研究了紧基上纤维$\mbb P^k$的全纯自同态的斜积映射。我们定义了纤维Fatou分量，并证明了它们是伪凸的和Kobayashi双曲的。
英文摘要：
In the first part of the thesis, we study some dynamical properties of skew products of Hénon maps of $\mbb C^2$ that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. Further, it is shown that the successive pullbacks of a suitable current by the skew Hénon maps converge to a multiple of the fibered stable current. Second part of the thesis generalizes most of the above-mentioned results for a completely random sequence of Hénon maps. In addition, for this random system of Hénon maps, we introduce the notion of average Green functions and average Green currents which carry many typical features of the classical Green functions and Green currents. Third part consists of some results about the global dynamics of a special class of skew maps. To prove these results, we use the knowledge of dynamical behavior of pseudo-random sequence of Hénon maps widely. We show that the global skew map is strongly mixing for a class of invariant measures and also provide a lower bound on the topological entropy of the skew product. We conclude the thesis by studying another class of maps which are skew products of holomorphic endomorphisms of $\mbb P^k$ fibered over a compact base. We define the fibered Fatou components and show that they are pseudoconvex and Kobayashi hyperbolic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
