Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”)

2021-10-16

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An intro to holomorphic dynamics, the study of iterated complex functions.

Video on Newton's fractal: /youtube/video/-RdOwhmqP5s

Special thanks to these supporters: https://3b1b.co/lessons/holomorphic-dynamics#thanks

Extra special thanks to Sergey Shemyakov, of Aix-Marseille University, for helpful conversations and for introducing me to this phenomenon.

Introduction to Fatou sets and Julia sets, including a discussion of Montel's theorem and its consequences:

http://www.math.stonybrook.edu/~scott/Papers/India/Fatou-Julia.pdf

Numberphile with Ben Sparks on the Mandelbrot set:

/youtube/video/FFftmWSzgmk

Ben explains how he made the Geogebra files on his channel here:

https://youtu.be/ICqj7nJbiRI (part 1)

https://youtu.be/3BXoNOahFJk (part 2)

Excellent article on Acko.net, from the basics of building up complex numbers to Julia sets.

https://acko.net/blog/how-to-fold-a-julia-fractal/

Bit of a side note, but if you want an exceedingly beautiful rendering of the quaternion-version of Julia fractals, take a look at this Inigo Quilez video:

https://www.youtube.com/watch?v=rQ2bnU4dkso

I first saw Fatou's theorem in this article:

https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-91/issue-2/On-the-iteration-of-a-rational-function--computer-experiments/cmp/1103940533.pdf

Moduli spaces of Newton maps:

https://arxiv.org/pdf/1512.05098.pdf

On Montel's theorem:

https://people.ucsc.edu/~fmonard/Sp17_Math207/lecture11.pdf

On Newton's Fractal:

https://core.ac.uk/download/pdf/1633889.pdf

Thanks to these viewers for their contributions to translations

Hebrew: Omer Tuchfeld

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These animations are largely made using a custom python library, manim. See the FAQ comments here:

https://www.3blue1brown.com/faq#manim

https://github.com/3b1b/manim

https://github.com/ManimCommunity/manim/

You can find code for specific videos and projects here:

https://github.com/3b1b/videos/

Music by Vincent Rubinetti.

https://www.vincentrubinetti.com/

Download the music on Bandcamp:

https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown

Stream the music on Spotify:

https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

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Timestamps:

0:00 - Intro

3:02 - Rational functions

4:15 - The Mandelbrot set

8:12 - Fixed points and stability

12:51 - Cycles

16:25 - Hidden Mandelbrot

21:17 - Fatou sets and Julia sets

26:24 - Final thoughts

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