Holomorphic Dynamics in Fatou Components in Two Complex Variables

Complex dynamics, also called holomorphic dynamics, is a branch of dynamical systems that intersects deeply with complex geometry and complex analysis. It investigates the behavior and evolution of a complex manifold M under the repeated application of a holomorphic function F : M → M . This thesis explores holomorphic dynamics from the prospective of the stable set, known as the Fatou set, and its connected components, called Fatou components. The work begins with an overview of key findings in one-dimensional holomorphic dynamics, then extends to higher-dimensional settings, with a particular focus on two-dimensional holomorphic dynamics. The novel contribution of this research lies in the construction of transcendental Hénon maps that exhibit cycles of escaping Fatou components with rank-1 limit functions. The most notable result is the fact that these Fatou components have disjoint and hyperbolic limit sets, offering a deeper understanding of the intricate and rich dynamics in higher dimensions.