Complex Dynamics

What happens when a simple formula on the complex numbers is applied over and over again?

4 chapters4 min read5 turning points 1 open problem

Branched from: Dynamical Systems + Complex Analysis
Branched into: Not yet surveyed past here
Figures: Gaston Julia, Pierre Fatou, Robert Brooks, J. Peter Matelski, Benoit Mandelbrot, Adrien Douady, John Hubbard, Dennis Sullivan, Mitsuhiro Shishikura

In brief

Complex dynamics studies what happens when a function of a complex number, as simple as $z \mapsto z^2 + c$ , is applied repeatedly. Some starting points stay bounded forever and others escape to infinity. The boundary between them, the Julia set, is usually a fractal of infinite intricacy. The Mandelbrot set catalogues, for every $c$ at once, which kind of Julia set appears.

Pierre Fatou and Gaston Julia founded the subject around 1918 using the tools of complex analysis, without being able to see what they described. It was largely dormant until computers drew the pictures in the late 1970s. Douady, Hubbard, Sullivan and others then turned the pictures into theorems, and the Mandelbrot set became one of the best-known images in mathematics. Whether it is locally connected is still unknown.

Key ideas

IterationEnters 1918 – 1920: Applying the same function again and again: $z_0$ , $f(z_0)$ , $f(f(z_0))$ , and so on. The sequence of results is the orbit of $z_0$ .
Julia set and Fatou setEnters 1918 – 1920: The Fatou set is where iteration behaves tamely, so nearby points stay nearby. The Julia set is the rest, where it is chaotic. For polynomials, the Julia set is the boundary of the set of points whose orbits stay bounded.
Mandelbrot setEnters 1982: The set of values $c$ for which the orbit of 0 under $z \mapsto z^2 + c$ stays bounded. It is exactly the set of $c$ whose Julia set is connected.
FractalEnters 1978 – 1980: A shape with detail at every scale, often with a fractional dimension. Mandelbrot coined the word in 1975, and Julia sets are among the richest examples.

Chapter I

Iteration Without Pictures

In 1918 the Paris Academy of Sciences offered its Grand Prix for work on iterating functions. The winner was Gaston Julia, aged twenty-five, who had lost his nose to a wound in the First World War and wore a leather patch for the rest of his life. Pierre Fatou had reached many of the same results independently, and the two quarrelled bitterly over priority. Between them they used the new theory of normal families from complex analysis to split the plane into a region where iteration is tame and a region where it is chaotic.

They could only imagine what these regions looked like. Calculating even one detailed picture by hand was out of the question. After a burst of activity, the subject lay mostly dormant for sixty years.

Chapter II

The Pictures Arrive

By the late 1970s computers could test millions of points. Robert Brooks and Peter Matelski printed a crude image in 1978 of the set of parameters $c$ for which $z^2 + c$ keeps 0 bounded. Benoit Mandelbrot, at IBM, drew it in detail and saw that it was extraordinary. Zooming in reveals spirals, filaments and tiny copies of the whole set, without end. Mandelbrot had coined the word "fractal" in 1975 for shapes with detail at every scale, and this became its emblem.

Then the pictures became theorems. Adrien Douady and John Hubbard proved in 1982 that the set is connected, although the pictures seemed to show separate islands, and named it after Mandelbrot. In 1985 Dennis Sullivan settled a question left open by Fatou, using methods borrowed from hyperbolic geometry.

Chapter III

A Closer Look: Bounded or Escaping?

Take the rule $z \mapsto z^2 + c$ , start from $z = 0$ and watch what happens for a few values of $c$ :

$c$	Orbit of 0	Fate
$0$	$0, 0, 0, \ldots$	fixed, bounded
$-1$	$-1, 0, -1, 0, \ldots$	cycle of length 2, bounded
$i$	$i,\ -1+i,\ -i,\ -1+i,\ -i, \ldots$	eventually a 2-cycle, bounded
$-2$	$-2, 2, 2, 2, \ldots$	fixed from then on, bounded
$1$	$1, 2, 5, 26, 677, \ldots$	escapes to infinity

For $c = i$ , squaring $-1 + i$ gives $(-1)^2 + 2(-1)(i) + i^2 = -2i$ , and adding $i$ gives $-i$ . Squaring $-i$ gives $-1$ , and adding $i$ gives $-1 + i$ again. The orbit is trapped.

The Mandelbrot set is the set of all $c$ whose orbit stays bounded, so $0$ , $-1$ , $i$ and $-2$ are in it and $1$ is not. There is a simple test: once an orbit gets farther than 2 from the origin, it must escape. That is how computers draw the set, colouring each $c$ by how many steps its orbit takes to pass distance 2.

The boundary is where it gets hard. The real number $c = 0.25$ is in the set: its orbit creeps up towards $0.5$ and never passes it. But $c = 0.26$ , only a hundredth larger, escapes after 30 steps, and $c = 0.251$ after 97. The closer $c$ is to the boundary, the longer it takes to decide. For points exactly on the boundary, no finite computation can settle the question. That is why the set's finest structure is still not fully understood.

Chapter IV

Still Unmapped

In 1998 Mitsuhiro Shishikura proved that the boundary of the Mandelbrot set is as complicated as a boundary in the plane can be, with dimension 2. The central conjecture of the field, that the set is locally connected, would give a complete description of it, and with it of all quadratic dynamical systems. Jean-Christophe Yoccoz proved it at most points, and his work contributed to his Fields Medal in 1994. The remaining points, deep inside nested copies of the set, are still out of reach. The same zooming, or renormalisation, that produces those copies also explains the universal constants of chaos theory.

Turning points5

1918 – 1920REFORMULATIONContested
Fatou and Julia found the theory of iteration
The Paris Academy offers its 1918 Grand Prix for work on iterating functions. Gaston Julia, who had lost his nose to a wound in the First World War, wins it with a long memoir, and Pierre Fatou publishes closely overlapping results. Using Montel's theory of normal families from complex analysis, they split the plane into regions of tame and chaotic behaviour. They could not draw what they had described.
What is disputed
Fatou and Julia reached many of the same results independently at the same time, and there was a bitter priority dispute between them, argued through notes to the Academy. Historians generally treat them as co-founders, and the two halves of the plane are named after one each.
Gaston Julia, Pierre Fatou
› Sources (2)
- Julia, G. (1918). Mémoire sur l'itération des fonctions rationnelles. Journal de Mathématiques Pures et Appliquées (8) 1: 47–245.
- Fatou, P. (1919–1920). Sur les équations fonctionnelles. Bulletin de la Société Mathématique de France 47: 161–271; 48: 33–94, 208–314.
1978 – 1980REFORMULATIONContested
The Mandelbrot set is drawn
With computers able to test millions of points, the set of parameters $c$ for which $z^2 + c$ keeps 0 bounded is drawn for the first time. Robert Brooks and Peter Matelski publish a crude picture in 1978. Benoit Mandelbrot, at IBM, publishes detailed studies in 1980. The images reveal an object of unexpected, endless complexity, and brought the field back to life.
What is disputed
Brooks and Matelski produced the first published picture, in a paper written in 1978, and some mathematicians thought the set should not carry Mandelbrot's name. Mandelbrot's detailed computer exploration, his recognition of its significance, and his publicity made it famous, and Douady and Hubbard named it after him.
Robert Brooks, J. Peter Matelski, Benoit Mandelbrot
› Sources (2)
- Mandelbrot, B. B. (1980). Fractal aspects of the iteration of z ↦ λz(1−z) for complex λ and z. Annals of the New York Academy of Sciences 357: 249–259.
- Brooks, R. & Matelski, J. P. (1981). The dynamics of 2-generator subgroups of PSL(2,C). In Riemann Surfaces and Related Topics, Annals of Mathematics Studies 97: 65–71.
1982PROOF
The Mandelbrot set is connected
Computer pictures showed apparently separate islands scattered around the Mandelbrot set. Adrien Douady and John Hubbard prove that the set is connected: every island is joined to the main body by filaments too thin to see. Their Orsay notes built the tools, external rays and parameter-space analysis, that the subject still uses.
Adrien Douady, John Hubbard
› Sources (1)
- Douady, A. & Hubbard, J. H. (1982). Itération des polynômes quadratiques complexes. Comptes Rendus de l'Académie des Sciences Paris 294: 123–126.
1985PROOF
No wandering domains
Fatou had asked whether a region of tame behaviour could wander forever without returning. Dennis Sullivan proves that for rational functions it cannot, by importing quasiconformal methods from the study of Kleinian groups. The proof completed the classification of tame behaviour that Fatou had begun, and showed the power of the dictionary between iteration and Kleinian groups.
Dennis Sullivan
› Sources (1)
- Sullivan, D. (1985). Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou–Julia problem on wandering domains. Annals of Mathematics 122(3): 401–418.
1998PROOF
The boundary has dimension two
Mitsuhiro Shishikura proves that the boundary of the Mandelbrot set, a curve-like set, is so wrinkled that its Hausdorff dimension is 2, as large as that of a filled region of the plane. It is as complicated as a boundary in the plane can possibly be.
Mitsuhiro Shishikura
› Sources (1)
- Shishikura, M. (1998). The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Annals of Mathematics 147(2): 225–267.

Applications

Where it is used

Numerical computation
Where Newton's method goes
Newton's method for finding roots is an iteration, and Cayley asked in 1879 which starting points lead to which root. Complex dynamics answers it: the basins are separated by fractal Julia sets. The theory also gives explicit starting points that are guaranteed to find every root of a polynomial.
› Sources (1)
- Hubbard, J., Schleicher, D. & Sutherland, S. (2001). How to find all roots of complex polynomials by Newton's method. Inventiones Mathematicae 146(1): 1–33.
Computer graphics
Fractal landscapes
Fractal geometry gave graphics a way to generate mountains, coastlines and clouds from a few rules. Loren Carpenter's fractal terrain in 1980 led to the first computer-generated planet sequence in a feature film, in Star Trek II (1982).
› Sources (1)
- Fournier, A., Fussell, D. & Carpenter, L. (1982). Computer rendering of stochastic models. Communications of the ACM 25(6): 371–384.
Universality↗ Physics · Phase Transitions
Why Feigenbaum's constant is universal
Physicists explained the universality of period doubling with a renormalisation argument, a zoom that turns a system into a smaller copy of itself. Mathematicians working in complex dynamics, notably Sullivan, McMullen and Lyubich, turned that argument into a proof, explaining numbers measured in fluids and circuits.
› Sources (1)
- Lyubich, M. (1999). Feigenbaum–Coullet–Tresser universality and Milnor's hairiness conjecture. Annals of Mathematics 149(2): 319–420.

Open problems

Where the map runs out

Open

Is the Mandelbrot set locally connected? (MLC)

Open as of 2026; proved at many parameter values, beginning with Yoccoz's work around 1990 and most recently at the Feigenbaum points (Dudko and Lyubich).

Douady and Hubbard conjectured that the Mandelbrot set is locally connected: near each of its points, the nearby parts are connected to each other, with no infinitely fine comb-like structure. If true, their theory gives a complete combinatorial description of the whole set.

Why it is hard: Local connectivity has to be proved at every point of the boundary, and different points need different techniques. Jean-Christophe Yoccoz proved it at all points except those lying in infinitely many nested smaller copies of the set, where renormalisation methods are needed and are only partly understood.
What resolving it unlocks: MLC implies the density of hyperbolicity for quadratic polynomials: every $c$ can be moved, by an arbitrarily small change, to a parameter with stable, predictable dynamics. It would complete the description of the simplest non-trivial family of dynamical systems.

› Sources (2)

Douady, A. & Hubbard, J. H. (1984–1985). Étude dynamique des polynômes complexes I–II. Publications Mathématiques d'Orsay.
Milnor, J. (2006). Dynamics in One Complex Variable (3rd ed.). Princeton University Press.

In brief

Key ideas

Iteration Without Pictures

The Pictures Arrive

A Closer Look: Bounded or Escaping?

Still Unmapped

Where it is used

Where Newton's method goes

Fractal landscapes

Why Feigenbaum's constant is universal

Where the map runs out

Is the Mandelbrot set locally connected? (MLC)

Further reading