Dynamical properties of families of holomorphic mapping

TL;DR

This thesis investigates the complex dynamics of skew products of Hénon maps, including pseudo-random and random sequences, and extends results to holomorphic endomorphisms of projective space, revealing properties like mixing, entropy, and hyperbolicity.

Contribution

It introduces fibered Green functions and currents for pseudo-random Hénon maps, generalizes these to random sequences, and studies global dynamics and properties of skew products of holomorphic endomorphisms.

Findings

Convergence of pullbacks to fibered stable currents

Introduction of average Green functions and currents for random systems

Proving strong mixing and entropy bounds for skew maps

Abstract

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.