Dynamical properties of families of holomorphic mapping

Ratna Pal

Published 2015-07-25Version 1

In the first part of the thesis, we study some dynamical properties of skew products of H\'enon 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 H\'enon mappings. In analogy with the dynamics of the iterates of a single H\'enon 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 H\'enon 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 H\'enon maps. In addition, for this random system of H\'enon 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 H\'enon 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.