Parameter Ray Atlas Tables

Douady-Hubbard Parameter Rays provide a theoretical foundation for discussing features of the M-set. In particular, rays allow us to characterize any particular location on the M-set by a single number, the ray arc angle. This web page provides tables and expressions for various interesting features on the Mandelbrot set. These tables should be used with the accompanying visual atlas, to visually locate the features given in the tables below.

Main Cardiod Buds

Buds on the main Cardiod can be labelled with Farey numbers. See, for example, Robert L. Devaney's 'How To Count' web page, which explains how to label the buds, and the 'How to Add' page which explains Farey addition (mirrored here).

Every bud on the main cardiod is pinched off by a pair of rays. The table below provides the ray angles for some of the buds on the main cardiod. In this table, p and q are the bud label, as per Devaney, and a1 and a2 are the angles of the two landing rays that pinch off that bud from each side.
p q a1 a2
1 2 1/3 2/3
1 3 1/7 2/7
1 4 1/15 2/15
1 5 1/31 2/31
1 ... ... ...
1 n 1/(2n-1) 2/(2n-1)
... ... ... ...
2 5 9/31 10/31
2 7 17/127 18/127
2 9 33/511 34/511
2 11 65/2047 66/2047
2 ... ... ...
2 2n+1 (2n+1+1)/(22n+1-1) (2n+1+2)/(22n+1-1)
... ... ... ...
3 7 41/127 42/127
3 8 73/255 74/255
3 10 145/1023 145/1023
3 11 273/2047 274/2047
3 13 545/8191 546/8191
3 14 1057/16383 1058/16383
3 ... ... ...
3 3n+1 (22n+1+2n+1+1)/(23n+1-1) (22n+1+2n+1+2)/(23n+1-1)
3 3n+2 (22n+2+2n+1+1)/(23n+2-1) (22n+2+2n+1+2)/(23n+2-1)
... ... ... ...
4 4n+1 (23n+1+22n+1+2n+1+1)/(24n+1-1) ...
4 4n+3 (23n+3+22n+2+2n+1+1)/(24n+3-1) ...

Symbolic Bud Codewords

The table presented above generalizes through the idea of 'codeword concatenation'. The basic idea is that for any given bud on the main cardiod, we can construct a binary string, a 'codeword', that is exactly equal to the ray angle. The codeword represents the periodicity of the points orbiting in that bud. Its best understood as a binary number because period-doubling inherently invites a binary/left-right representation. So lets explain what we mean by this.

The 'main sequence' buds on the main cardiod have periodicity n and have Farey labels 1/n. We define the codeword for these buds to be n-1 zeros followed by 1. Its length is of course n binary digits or bits. If we put a decimal point in front of the codeword, it would equal 1/2n. If we made the codeword repeat itself an infinite number of times, then its value would be 1/(2n-1). The Bernoulli map of this codeword has period n, the same as the period of the bud. And, of course, the smaller of the two ray angles for this bud is just the value of the repeated codeword, viz. 1/(2n-1).

The codewords for other buds are given by concatenating the codewords of their Farey parents, with the left parent contributing the codeword on the right, and the right parent the left codeword. Thus, the codeword for 2/(2n+1) = 1/n {+} 1/(n+1) is (n-1) zeros, 1, n zeros,1, or expressed as a fraction, (2n+1+1)/22n+1. The length of this codeword is of course, (2n+1), the same as the Farey fraction denominator (which is the bud periodicity). The ray angle is given by repeating the codeword over and over, viz. (2n+1+1)/(22n+1-1). It is not hard to verify that by concatenating codewords, we can get every other value in the table above.

Of course, this begs the question, what features are other codewords associated with? The answer is, of course, the tails.

Into But Not Onto

The main bud can be mapped into any smaller bud by performing a binary substitution for the tail values (who first noted this?). Furthermore, this mapping seems 'morphic' in some way, in that it preserves features: for any feature on the main bulb, the mapping provides the corresponding feature on the mini-bulb.

Thus, for example, the tail of the bud at (re,im)=(-1.75,0) is rayed by 3/7 and 4/7. Now pick a number, any number, that points at an interesting feature on the main bulb. Say its binary expansion is 0.b1b2b3b4... Create a new binary expansion by concatenating the two strings (011) and (100). Whenever bk is zero, use (011), and if its one, use (100). Thus, for example, on the main bulb, the ray 1/3 = 0.0101010101... pinches off the west bud. Substituting, we get 0.011100011100011100 .. = 4/9. Indeed, as the picture shows, that's where the 4/9th's ray goes.

Generally, the two rays that point at the tail-end are given by p/(2n-1) and q/(2n-1) for some p, q and n. One the main sequence, we have q=p+1. The two codewords for the above substitution are then p/2n for 0 and q/2n for 1.

Farey Numbers

Farey Numbers are important to the study of chaos because they appear where ever there is period doubling. This is in part because whenever one has period doubling, one has a binary tree, whose branches are labelled by binary fractions. The Farey number mapping simply takes Farey numbers, expressed as binary numbers, into the corresponding continued fractions. Thus, for example, the binary sequence for 1/7th is 0.001001001001001... We can write the equivalent continued fractions as [0;3,1,2,1,2,1,2,1,2,1,...] whose value is 1/(2+sqrt(3)).

More generally, we have 1/(2n-1) whose continued fraction is [0;n,1,(n-1),1,(n-1),1,(-1),1,...]. The value of this continued fraction is 2/(n+1+sqrt(n2+2n-3)). The table below gives some other relationships.

Farey Number Expansion Value Notes
1/(2n-1) [0;n,1,(n-1),1,(n-1),1,(n-1),1,...] 2/(n+1+sqrt(n2+2n-3))
2/(2n-1) [0;(n-1),1,(n-1),1,(n-1),1,(n-1),1,...] 2/(n-1+sqrt(n2+2n-3))
2m/(2n-1) [0;(n-m),1,(n-1),1,(n-1),1,(n-1),1,...] 2/(n-2m+1+sqrt(n2+2n-3)) m < n
1/n [0;n] 1/2n

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Linas Vepstas December 2000
linas@linas.org
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