External rays

Dynamics of the map fo(z) = z2 is very simple. Remind that in polar coordinate z = re2πit one get rn+1 = rn2 and tn+1 = 2tn mod 1. The latter is the well known sawtooth map.

Theorem of Böttcher

For large |z| one can neglect the c value in the map fc(z) = z2 + c therefore it is close to the map z2. This equivalence extends up to the connected Julia set border (and the unit circle correspondingly). In accordance with the Böttcher theorem [1] there is a unique analytic isomorphism w = φ(z) mapping f into z2 or φ o f o φ-1(w) = w2. At infinity φ(z) tends asymptotically to the identical map.

To compute |φ(z)| at a point z we must simply follow the orbit zk = fcok(z) until at a step k we reach some point which is very far from 0 where fc is close to fo (and zk to wk). Then we make k itterations fo-1 (in the "reverse" direction) i.e.
    |φ(z)| = limk→∞ |f ok(z)|1/2k.

Green's function and external rays for Julia sets

It is well known in 2D electrostatics that if there is an analytic function w = φ(z) mapping the exterior of the Julia set to the exterior of the unit circle then the potential or Green's function of the "charged" Julia set at a point z is equal to the potential of the charged unit circle G = log|w| (at the point w) or
    G(z) = limn→∞ ln|f on(z)| / 2n.
This implies also that
    G(f(z)) = 2 G(z).
It is evident that far from charged body equipotential curves are circles. Let R is a large circle then
    G(f -ok(R)) = G(R)/2k = const
therefore preimages of the circle are equipotential curves for the Julia set (see the picture to the left for R = 10).
Electric field lines (orthogonal to equipotential curves) are called external rays. They are preimages of the field lines of charged circle r exp(2πit), r>1. Each such ray Rt can be specified by its angle at infinity t. Due to isomorphism φ(z) external rays dynamics under f is the same as dynamics of the straight rays under z2. E.g.
    f(Rt ) = R2t .
To the left every point is painted in the red, green and blue colors if for r > 10 its orbit falls into the sectors (0o, 120o), (120o, 240o), (240o, 360o) correspondigly. With enough patience you can see equipotential curves and orthogonal field lines. Exteral rays with rational angles t = 0, 1/4, 1/3, 1/2... are marked here.
At last to the left external rays p/12 for p = 0,2,..11 are plotted by means of the Wolf Jung's algorithm. As since I used only simplest Jung's magic this applet is a bit buggy for large q. If you need flexible and accurate program to explore external rays and much more use his Mandel.

I am grateful to Adam Majewski for he convinced me to translate Wolf Jung's sources into Java.

How to calculate external rays

For the quadratic map f on = [f o(n-1)]2 + c there is identity
f on = (f o(n-1))2 [f on/(f o(n-1))2] = (f o(n-1))2 [f on/(f on - c)].
Therefore one get recursively
φc(z) = limn→∞ [fcon(z)]1/2n = z ∏n=1,∞ [fcon(z)/(fcon(z)-c)]1/2n
and for external angles correspondingly
argc(z) = arg(z) + ∑n=1,∞1/2n arg(fcon(z)/(fcon(z)-c)).
Wolf Jung used this formula to calculate external angles with a little "Voodoo" when
|arg(fcon(z)/(fcon(z)-c))| > π.

External rays for the Mandelbrot set

If Julia set is totally disconnected then the value G(0) = G(c)/2 > 0 plays a special role. There is a canonical conformal isomorphism w = ψc(z) from the open set G(z) > G(0) to the region log|w|>G(0). The map z → f(z) on this region is conjugate under ψc to the map w → w2, and the equipotentials and dynamic rays in the z-plane correspond to concentric circles and straight half-lines through the origin respectively in the w-plane.
In particular, if we choose a constant Go > G(0), then G(z) = Go is a simple closed curve, canonically parametrized by the angle of the corresponding dynamic ray. In particular, the critical value z = fc(0) = c has a well defined external angle, which we denote by t(c). However, for Go = G(0) this points G(z) = G(0) make a figure eight curve (see to the left). The open set G(z) < G(0) splits as a disjoint union Uo and U1, where the Ub are the regions enclosed by the two lobes of this figure eight.
According to Douady and Hubbard ψc is a conformal isomorphism from the complement of M onto the complement of the closed unit disk. Therefore it determines equipotential curves and external rays for the Mandelbrot set.

It is known that an external ray whose angle is rational actually lands on M. That is
    lim r→1 ψ-1(re2πit)
exists and is a unique point on the boundary of M. For example, the ray with angle 0 lies on the real axis and lands on M at the cusp of the main cardioid, namely C = 1/4. Also, the ray with angle 1/2 lies on the negative real axis and lands on M at the tip of the tail of M at C = -2.

[1] J. Milnor Dynamics in One Complex Variable: Introductory Lectures. Preprint ims90-5

Contents     Previous:     Next: Periodic orbit and external rays
updated 19 Mar 08