Julia set

[a] These sets are named after the French mathematicians Gaston Julia[1] and Pierre Fatou[2] whose work began the study of complex dynamics during the early 20th century.

and are such that: The last statement means that the termini of the sequences of iterations generated by the points of

The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

[3] Each component of the Fatou set of a rational map can be classified into one of four different classes.

the Julia set is the unit circle and on this the iteration is given by doubling of angles (an operation that is chaotic on the points whose argument is not a rational fraction of

There are two Fatou domains: the interior and the exterior of the circle, with iteration towards 0 and ∞, respectively.

For such an iteration the Julia set is not in general a simple curve, but is a fractal, and for some values of c it can take surprising shapes.

This is because of the following result on the iterations of a rational function: Theorem — Each of the Fatou domains has the same boundary, which consequently is the Julia set.

This phenomenon happens, for instance, when f(z) is the Newton iteration for solving the equation

) Then the filled Julia set for this system is the subset of the complex plane given by where

[6] The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Adam Epstein's finite-type maps.

Julia sets are also commonly defined in the study of dynamics in several complex variables.

Consider implementing complex number operations to allow for more dynamic and reusable code.

This can be implemented, very simply, like so: The difference is shown below with a Julia set defined as

, it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that

[8] This means that the potential function on the outer Fatou domain defined by this correspondence is given by: This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula.

, which should be regarded as the real iteration number, and we have that: If the attraction is ∞, meaning that the cycle is super-attracting, meaning again that one of the points of the cycle is a critical point, we must replace α by where w′ is w iterated r times and the formula for φ(z) by: And now the real iteration number is given by: For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H−1 (H = 500, for instance).

by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H. The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral.

, as defined in the previous section), the bands of iteration show the course of the equipotential lines.

), we can easily show the course of the field lines, namely by altering the colour according as the last point in the sequence of iteration is above or below the x-axis (first picture), but in this case (more precisely: when the Fatou domain is super-attracting) we cannot draw the field lines coherently - at least not by the method we describe here.

(the r-fold composition), and we define the complex number α by If the points of C are

has (in connection with field lines) character of a rotation with the argument β of α (that is,

In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration

A colouring of the field lines of the Fatou domain means that we colour the spaces between pairs of field lines: we choose a number of regularly situated directions issuing from

A coloured field line (the domain between two field lines) is divided up by the iteration bands, and such a part can be put into a one-to-one correspondence with the unit square: the one coordinate is (calculated from) the distance from one of the bounding field lines, the other is (calculated from) the distance from the inner of the bounding iteration bands (this number is the non-integral part of the real iteration number).

of f. Unfortunately, as the number of iterated pre-images grows exponentially, this is not feasible computationally.

As a Julia set is infinitely thin we cannot draw it effectively by backwards iteration from the pixels.

where p(z) and q(z) are complex polynomials of degrees m and n, respectively, and we have to find the derivative of the above expressions for φ(z).

⁠ and having order r, we have and consequently: For a super-attracting cycle, the formula is: We calculate this number when the iteration stops.

Besides drawing of the boundary, the distance function can be introduced as a 3rd dimension to create a solid fractal landscape.