Fractals

Mandelbrot set

The Mandelbrot set is the set complex numbers for which the function

does not diverge when iterated from , i.e. remains bounded in absolute value.

Formally, the Mandelbrot set is the set of values of in the complex plane for which the orbit of under iteration of the quadratic map

remains bounded.

The Mandelbrot set is defined by a family of complex quadratic polynomials

given by

where is a complex parameter. For each , one considers the behavior of the sequence

obtained by iteration starting at critical point . This either escapes to infinity or stays within a disk of finite radius; the Mandelbrot set is then the set of all points such that the sequence stays within a disk of finite radius.

More formally, if denotes the n-th iterate of (i.e. composed with itself times), the Mandelbrot set is the subset of the complex plane given by

As explained below, it is in fact possible to simplify this definition by taking .

Visualizations of Mandelbrot sets

• Color all points that belong to the Mandelbrot set black and all other points white

Connection to Julia sets

An equivalent definition of the Mandelbrot set is as the subset of the complex plane consisting of those parameters for which the Julia set of is connected.

Julia set

Let be a holomorphic function from the Riemann sphere to itself.

Such are precisely the complex rational functions, that is

where and are complex polynomials.

Assume that and are non-constant, and that they have no common roots (i.e. does not divide ). Then there is a finite number of open sets , that are left invariant by and are such that:

1. is dense in the plane
2. behaves in a regular and equal way on each of the sets

The last statement means that the termini of the sequences of iterations generated by the points of are either

• the same set => finite cycle
• are finite cycles of circular or annular shaped sets centered at the same point

These sets are the Fatou domains of , and their union is the Fatou set of .

Further, each of the Fatou domains contains at least on critical point of , i.e. a (finite) point such that

if either of the following is the case:

• degree of the numerator is at least two larger than the degree of the denominator
• for some and a rational function satisfying this condition

The complement of the Fatou set is the Julia set of .

is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers).

Like , is left invariant by , and on this set the iteration is repelling, meaning that

for all in a neighborhood of (within ).

This means that behaves chaotically on the Julia set.

There are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and thus they only make up a infinitely small part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

Equivalent definitions of Julia sets are:

• is the smallest closed set containing at least three points which is completely invariant under (Möbius transformation?)
• is the closure of the set of repelling periodic points
• For all but at most two points , the Julia set is the set of limit points of the full backwards orbit

i.e. limit points of the union of pre-images of all the iterations.

• If is an entire function, then is the boundary of the set of points which converge to infinity under iteration.
• If is a polynomial, then is the boundary of the filled Julia set: i.e. those points whose orbits under iterations of remain bounded.

Julia set of a Mandelbrot set

Mandelbrot sets are given by

where is a complex parameter.

The Julia set of for this system is the subset of complex plane given by:

where is the n-th iterator of , i.e. , n times.