Theorem 1. The Critical Value Sector S_{1} [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 S_{1} based at some point z_{1} which
contains the critical value c = f(0), and whose closure contains no
point other than z_{1} of the orbit O. This critical
value sector is the unique sector of smallest angular width.
For example, you see above period-3 orbit {p_{1},
p_{2}, p_{3}} with portrait
P_{A} = {{10/63, 17/63}, {20/63, 34/63}, {5/63, 40/63}} and the fixed point z_{2} with portrait P_{B} = {{1/7, 2/7, 4/7}} . It is convenient to represent such portraits by a schematic diagram, as shown in Fig.1 |
To the left you see W_{A} and W_{B} 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 W_{p} 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 W_{B} wake there is the fixed point z_{2} with portrait P_{B} and v = 3. You see below that unstable period-3 orbit with valence one degenerates to form period-1 orbit (the fixed point z_{2}) with portrait P_{B} when C crosses the root point of the 1/3 bulb. |
Note that J-ray with angle 0 always lands at the right unstable fixed
point z_{1} and J-ray with angle 1/2 lands at symmetrical
preimage of this point -z_{1}.
E.g. for period-3 unstable orbit {p_{1}, p_{2}, p_{3}} 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