Holomorphic dynamics studies the evolution of complex manifolds under the iteration of holomorphic maps.

While significant progress has been made in understanding the theory of one-dimensional holomorphic dynamics, the transition to higher dimensions still presents difficult challenges since the situation is vastly different from the one-dimensional case.

Even only the study of the dynamics of automorphisms (i.e. holomorphic maps injective and surjective) in two dimensions already poses deep difficulties, and the construction of significant examples is an active area of research.

In this talk, we provide an overview of the dynamics in several complex variables, focusing particularly on the stable dynamics of automorphisms of C^2. We introduce concepts such as Fatou sets, polynomial and transcendental Hénon maps, and limit functions. Finally, we address two recently resolved questions that refer to the current state of my research  (a joint work with A. M. Benini and A. Saracco):

Can limit sets for (non-recurrent) Fatou components be hyperbolic?

Can limit sets be distinct?