Complex Dynamics of Transcendental Skew-Products

In complex dynamics, one studies the iteration of a holomorphic function, with a special focus on stable and chaotic behavior. Historically, functions from the complex plane to itself have been of particular interest. In this one-dimensional setting, there is an exhaustive characterization of stable components, known as Fatou components. In this talk, I will first give a brief overview of this classification before describing one approach to extending this understanding to higher dimensions, where similar characterizations have remained elusive. Specifically, I focus on skew-products, a simpler family of two-dimensional holomorphic functions.