Complex dynamics
Julia sets
For each complex parameter c, the Julia set is the boundary between starting points z whose iteration z ← z² + cstays bounded and points that escape to infinity. Studied by Gaston Julia and Pierre Fatou in 1917-1919, decades before Mandelbrot made them visual.
At a glance
| First studied | 1917-1919 (Julia and Fatou, independently) |
|---|---|
| Iteration | zn+1 = zn2 + c (c is the parameter; z varies) |
| Filled Julia set K(c) | {z : the orbit stays bounded} |
| Julia set J(c) | ∂K(c), the boundary |
| Fatou-Julia theorem | J(c) is connected iff c is in the Mandelbrot set |
Relation to Mandelbrot
The Mandelbrot set serves as a parameter atlas for Julia sets. For each c:
- If c is in the Mandelbrot set, the Julia set J(c) is connected.
- If c is outside, J(c) is a totally disconnected Cantor dust.
- If c is on the boundary (a Misiurewicz point), the Julia set is locally similar to the Mandelbrot boundary near c.
Famous c values
- c = 0: J = unit circle. The trivial case.
- c = -1: “basilica”, a chain of disks.
- c = -0.123 + 0.745i: Douady rabbit, three-fold rotationally symmetric ears.
- c = 0.285 + 0.01i: “San Marco” dragon.
- c = -0.4 + 0.6i: a delicate dendrite.
- c = -0.835 - 0.2321i: spirals everywhere.
Dimension
Julia sets for generic c on the Mandelbrot boundary have Hausdorff dimension that ranges over (1, 2). Specific values are calculable via Bowen’s formula relating dimension to the pressure of the dynamical system. Some Julia sets have Hausdorff dimension 2.
References
- Julia, G., “Mémoire sur l’itération des fonctions rationnelles,” J. Math. Pures Appl., 1918.
- Fatou, P., “Sur les équations fonctionnelles,” Bull. Soc. Math. France, 1919-1920.
- Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Westview, 2003.
- Carleson, L. & Gamelin, T. W., Complex Dynamics, Springer, 1993.
- Mandelbrot set · Julia explorer
Quick quiz
Test yourself on julia
10 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.For a fixed c, the filled Julia set K_c consists of:
Q2.If c lies inside the Mandelbrot set, the Julia set J_c is typically:
Q3.Which parameter c gives the famous 'Douady rabbit' Julia set?
Q4.For c = 0 the Julia set is:
Q5.If c is outside the Mandelbrot set, the Julia set is:
Q6.The Julia set is invariant under which kind of map?
Q7.The basilica Julia set corresponds approximately to:
Q8.Which is true for any non-empty Julia set J_c?
Q9.Iterating backward from any non-exceptional point inside the Julia set...
Q10.If c is a parabolic parameter (e.g., c on the boundary of the cardioid), the Julia set is...