# Dynamics of transcendental H\'enon maps

## TL;DR

This paper introduces and analyzes the dynamics of transcendental Hénon maps in the complex plane, focusing on the classification of Fatou components and identifying examples of Baker and wandering domains.

## Contribution

It extends the theory of polynomial Hénon maps to transcendental maps, providing a classification of Fatou components and new examples of dynamical behaviors.

## Key findings

- Recurrent invariant Fatou components are classified similarly to polynomial Hénon maps.
- Existence of Baker domains in transcendental Hénon maps.
- Existence of wandering domains in transcendental Hénon maps.

## Abstract

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex H\'enon maps. We introduce the family of transcendental H\'enon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial H\'enon maps, and we give examples of Baker domains and wandering domains.