information:classification
Field lines

In each Fatou domain (that is not neutral) there are two systems of lines orthogonal to each other: the equipotential lines (for the potential function or the real iteration number) and the field lines.

If we colour the Fatou domain according to the iteration number (and not the real iteration number ν (z) {\displaystyle \nu (z)}, as defined in the previous section), the bands of iteration show the course of the equipotential lines. If the iteration is towards ∞ (as is the case with the outer Fatou domain for the usual iteration z 2 + c {\displaystyle z^{2}+c}), 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. In this case a field line is also called an external ray.

Let z be a point in the attracting Fatou domain. If we iterate z a large number of times, the terminus of the sequence of iteration is a finite cycle C, and the Fatou domain is (by definition) the set of points whose sequence of iteration converges towards C. The field lines issue from the points of C and from the (infinite number of) points that iterate into a point of C. And they end on the Julia set in points that are non-chaotic (that is, generating a finite cycle). Let r be the order of the cycle C (its number of points) and let z ∗ {\displaystyle z^{*}} be a point in C. We have f (f (… f (z ∗))) = z ∗ {\displaystyle f(f(\dots f(z^{*})))=z^{*}} (the r-fold composition), and we define the complex number α by

If the points of C are z i, i = 1, …, r (z 1 = z ∗) {\displaystyle z_{i},i=1,\dots,r(z_{1}=z^{*})}, α is the product of the r numbers f ′ (z i) {\displaystyle f'(z_{i})}. The real number 1/|α| is the attraction of the cycle, and our assumption that the cycle is neither neutral nor super-attracting, means that 1 < 1 / | α | < ∞. The point z ∗ {\displaystyle z^{*}} is a fixed point for f (f (… f (z))) {\displaystyle f(f(\dots f(z)))}, and near this point the map f (f (… f (z))) {\displaystyle f(f(\dots f(z)))} has (in connection with field lines) character of a rotation with the argument β of α (that is, α = | α | e β i {\displaystyle \alpha =|\alpha |e^{\beta i}}).

In order to colour the Fatou domain, we have chosen a small number ε and set the sequences of iteration z k (k = 0, 1, 2, …, z 0 = z) {\displaystyle z_{k}(k=0,1,2,\dots,z_{0}=z)} to stop when | z k − z ∗ | < ϵ {\displaystyle |z_{k}-z^{*}|<\epsilon }, and we colour the point z according to the number k (or the real iteration number, if we prefer a smooth colouring). If we choose a direction from z ∗ {\displaystyle z^{*}} given by an angle θ, the field line issuing from z ∗ {\displaystyle z^{*}} in this direction consists of the points z such that the argument ψ of the number z k − z ∗ {\displaystyle z_{k}-z^{*}} satisfies the condition that

For if we pass an iteration band in the direction of the field lines (and away from the cycle), the iteration number k is increased by 1 and the number ψ is increased by β, therefore the number ψ − k β mod π {\displaystyle \psi -k\beta \mod \pi } is constant along the field line.

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 z ∗ {\displaystyle z^{*}}, and in each of these directions we choose two directions around this direction. As it can happen that the two field lines of a pair do not end in the same point of the Julia set, our coloured field lines can ramify (endlessly) in their way towards the Julia set. We can colour on the basis of the distance to the center line of the field line, and we can mix this colouring with the usual colouring. Such pictures can be very decorative (second picture).

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). Therefore, we can put pictures into the field lines (third picture).