---
tag: CNX4KVMDJQ
kind: concept
name: External ray
msc: [37F20, 37F10]
sources:
  - url: https://en.wikipedia.org/wiki/External_ray
same_as:
  wikidata: Q5422043
  wikipedia: External ray
---

## Summary

An external ray is a curve that comes in from infinity towards the filled Julia set of a polynomial, or towards the [[Mandelbrot set|QSWSVYRCS1]].
It is the image of a radial line under the inverse of the Böttcher coordinate, and it is labelled by an angle.
The polynomial maps each ray to another ray, multiplying its angle by the degree.
Rays and the points where they land turn the topology of [[Julia sets|5M4S7P04GR]] into combinatorics of angles.

## Explanations

### L1 human

Outside the filled Julia set of a polynomial with connected Julia set, there is a coordinate in which everything looks like the outside of the unit disc.
In that picture, draw the straight rays that leave the disc at each angle.
Pulled back to the original plane, these become curves that spiral and bend towards the Julia set.
For $z \mapsto z^2$, the rays are just the straight half-lines from the unit circle to infinity.
Squaring doubles the angle of each ray, and this simple rule organises the whole dynamics.