Periodic orbits and external rays

A key point in Douady and Hubbard's study of the Mandelbrot set is the theorem that every parabolic point c ≠ 1/4 in M is the landing point for exactly two external rays with angles which are periodic under doubling. By definition, a parameter point is parabolic if and only if the corresponding quadratic map has a periodic orbit with some root of unity as multiplier. It is the root point of a bulb. Thus the angles of the external rays determine the ordering of the bulbs in M

Theorem 1. The Critical Value Sector S1 [1]
Let O be an orbit of period p ≥ 1. If there are v ≥ 2 dynamic rays landing at each point of O, then there is one and only one sector S1 based at some point z1 which contains the critical value c = f(0), and whose closure contains no point other than z1 of the orbit O. This critical value sector is the unique sector of smallest angular width.

Periodic orbit portraits

Let O = {z1,...,zp} be a periodic orbit. Suppose that there is some rational angle t so that the dynamic ray Rt lands at a point of O. Then for each zi the collection Ai consisting of all angles of dynamic rays which land at the point is a finite and non-vacuous subset. The collection {A1,...,Ap} will be called the orbit portrait P = P(O). The number of elements in each Ai (or in other words the number of J-rays which land on each orbit point) will be called the valence v.
portrait For example, you see above period-3 orbit {p1, p2, p3} with portrait
  PA = {{10/63, 17/63}, {20/63, 34/63}, {5/63, 40/63}}
and the fixed point z2 with portrait
  PB = {{1/7, 2/7, 4/7}} .
It is convenient to represent such portraits by a schematic diagram, as shown in Fig.1

Wakes

Let 0 < t- < t+ < 1 be the angles of the two dynamic rays RJ which bound the critical value sector S1.
Theorem. The Wake Wp . The two corresponding parameter rays RM land at a single point ρp of the parameter plane. These rays cut the open subset Wp. A quadratic map fc has a repelling orbit with portrait P if and only if C is in Wp and has a parabolic orbit with portrait P if and only if C = ρp. Wp is called P wake.
To the left you see WA and WB wakes bounded by the 10/63, 17/63 and 1/7, 2/7 rays correspondingly.

As we follow a path in parameter space which crosses into the wake Wp through its root point, either one orbit with a portrait of valence one degenerates to form an orbit of lower period with portrait P, or else two different orbits with portraits of valence one fuse together to form an orbit with portrait P.

E.g. within the WB wake there is the fixed point z2 with portrait PB and v = 3. You see below that unstable period-3 orbit with valence one degenerates to form period-1 orbit (the fixed point z2) with portrait PB when C crosses the root point of the 1/3 bulb.

If we cross into Wp through a parameter ray RM, the picture is similar except that the landing point of the dynamic ray RJ jumps discontinuously. If t+ and t- belong to the same cycle under angle doubling, then the landing points of both of these dynamics rays jump discontinuously.
To the left above within WB three rays land at z2 (sorry the Cantor-like Julia set is invisible). When C quits the wake these rays jump and land at three different points.

Counting external angles

Remind that for angle doubling map we can read binary expansion of t by watch the orbit of t under doubling. We assign 0 to the itinerary if t get into the upper semicircle 0 ≤ t < 1/2 and 1 if t get into the lower semicircle 1/2 ≤ t < 1. Due to isomorphism φ(z) dynamics on connected Julia set under fc is equivalent to period doubling dynamics on unit circle. Therefore we can read itinerary for the landing point of an external J-ray in a similar way.
Note that J-ray with angle 0 always lands at the right unstable fixed point z1 and J-ray with angle 1/2 lands at symmetrical preimage of this point -z1.

E.g. for period-3 unstable orbit {p1, p2, p3} to the left three periodic iteneraries and corresponding external rays are 0.(001) = 1/7, 0.(010) = 2/7 and 0.(100) = 4/7.

In a similar way you can see that period-6 orbit below has itinerary 0.(001010) = 10/63 and corresponding exteral J-ray. The orbit starts at the center of the white square. The square is mapped into region with inverse color. Press the left button and drag by mouse the white square in series along the orbit to watch this itenerary.

[1] John W. Milnor Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account arXiv:math/9905169
[2] Wolf Jung Homeomorphisms on Edges of the Mandelbrot Set Ph.D. thesis of 2002
[3] Dierk Schleicher Rational parameter rays of the Mandelbrot set arXiv:math.DS/9711213


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updated 31 Mar 08