---
tag: 6WCRMYEYF9
kind: concept
name: Holomorphic dynamics in several variables
msc: [37F80, 32H50]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_dynamics
---

## Summary

Holomorphic dynamics in several variables studies the iteration of holomorphic maps $f \colon X \to X$ of a [[complex manifold|JSGWAN2RJV]] $X$ of dimension at least $2$.
The two main families of examples are holomorphic endomorphisms of complex projective space $\mathbb{P}^k$ and polynomial automorphisms of $\C^2$, such as the [[Hénon maps|KS4FJ6H5JH]].
As in one-variable [[complex dynamics|BQQZ0ZCGQF]], space splits into a Fatou set, where the dynamics is stable, and a Julia set, where it is chaotic.
Many tools are new, notably plurisubharmonic functions, positive closed currents and invariant measures.

## Explanations

### L1 human

Take a holomorphic map of a space to itself and apply it again and again.
Each starting point traces out an orbit, and the question is how these orbits behave.
In one complex variable this gives Julia sets and the Mandelbrot set.
In two or more variables the same question has richer answers: orbits can be attracted to curves, strange sets appear, and maps can be invertible yet chaotic.
The map $(x, y) \mapsto (x^2 + c - y, x)$ of $\C^2$ is a simple example: it is invertible, and its orbits already show this rich behaviour.