Enumeration of Features

Robert P. Munafo, 2010 April 5.

This is a description of how to determine the number of each different type of feature in the Mandelbrot Set.

The following formulas are fairly easy to prove if the following two properties have already been established:

The Farey addition property : if two mu-atoms are at internal angles a/b and c/d, with b relatively prime to d, and if the lowest-period mu-atom between them is e/f, and if e<a and e<c, then a+c=e and b+d=f. For a formal description of this, see the link in the Robert Devaney article.

Period scaling : The mu-unit belonging to a hyperbolic component of period p possesses a full set of higher period components of periods that are all multiples of p. This is mentioned as property 2.3 in the 1986 article by John Milnor.

The rest of the current discussion assumes the above properties as truth.

Mu-Atoms

We start by considering the lemniscates as polynomials set equal to zero, and solving them, i.e. "finding the roots". Each root gives the nucleus of some mu-atom. Since the nth lemniscate is of order 2^n-1, it follows that for each period n there are 2^n-1 roots.

However, a root of period 3 (for example) also shows up as a root of period 6, 9, etc. but we don't count it that way. It only counts as a root of period 3. So, from 2^n-1 we must subtract the number of roots of all lower periods that are divisors of n. This gives the following formula for the number of Mu-Atoms of period n:

N_a(n) = 2^n-1 - Σ_{(f:(f<n) and (n≡0 mod f))}N_a(f)

where n≡0 mod f means "f is a divisor of n". The extra condition f < n is there to exclude f equal to n.

For example, when computing N₆ we start with 2⁵, and then subtract the values of N₁, N₂ and N₃ because 1, 2 and 3 are divisors of 6.

Here are the first few values of the sequence:

N₁=1, N₂=1, N₃=3, N₄=6, N₅=15, N₆=27, N₇=63, N₈=120, N₉=252, N₁₀=495, N₁₁=1023, N₁₂=2010, N₁₃=4095, N₁₄=8127, N₁₅=16365, N₁₆=32640, N₁₇=65535, N₁₈=130788, N₁₉=262143, N₂₀=523770, N₂₁=1048509, N₂₂=2096127, N₂₃=4194303, N₂₄=8386440, N₂₅=16777200, N₂₆=33550335, N₂₇=67108608, N₂₈=134209530, N₂₉=268435455, N₃₀=536854005, N₃₁=1073741823, ...

(This is Sloane's sequence A000740.)

Secondary Continental Mu-Atoms

There is one secondary continental mu-atom for each rational number between 0 and 1. The secondary continental mu-atoms of period n correspond to the rational numbers with n in the denominator. Because of this, the number of secondary continental mu-atoms is equal to Euler's Totient function:

Φ(n) = n - Σ_{(f:gcd(n,f)=1)}1

That is, take n and subtract 1 for every number that is relatively prime to n. This sequence is Sloane's sequence A000010. It starts:

0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, ...

Because of period scaling, each mu-atom has children that have the same distribution as the secondary continental mu-atoms, but with scaled-up periods. From this, we can derive the formula for the number of continental mu-atoms of period n:

N_c(1) = 1 (special case)

N_c(n) = Σ_{(f:n≡0 mod f)} [N_c(f) ^. Φ(n/f)]

This sequence (Sloane's A006874) starts:

1, 1, 2, 3, 4, 6, 6, 9, 10, 12, 10, 22, 12, 18, 24, 27, 16, 38, 18, 44, 36, 30, 22, 78, 36, 36, 50, 66, 28, 104, 30, ...

The Islands

Next we consider the number of island mu-molecules. So far we have a total number of mu-atoms, and a number of continental mu-atoms, for each period. All mu-atoms that are not continental are part of an island.

However, some of these mu-atoms are descendants of the seeds of the island mu-molecules. So, first we have to look at the relation between descendants in general (which includes continental descendants) and mu-molecule seeds (including the continent seed).

The number of mu-molecules is equal to the number of seeds, since each mu-molecule has one seed. Furthermore, every mu-atom is either a seed or a descendant, but not both. Therefore, we have this relation between mu-molecules and descendants:

N_m(n) = N_a(n) - N_d(n)

Descendants

Number of descendants:

N_d(n) = Σ_{(f:n≡0 mod f)} [N_a(f)^.Φ(n/f)]

The sequence is Sloane's A006875, and starts:

0, 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, 324, 18, 561, 156, 1043, 22, 2340, 80, 4119, 540, 8307, 28, 17521, 30, ...

Mu-Molecules

From this we get the values of N_m(n), which is Sloane's sequence A006876:

1, 0, 1, 3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, 16315, 32496, 65519, 130464, 262125, 523209, 1048353, 2095084, 4194281, 8384100, 16777120, 33546216, 67108068, 134201223, 268435427, 536836484, 1073741793, ...

The functions in maxima

Here are each of the functions implemented in maxima.

Na(p) := block( [a:2^(p-1),f:0], for f in divisors(p) do if p>f then a : a - Na(f), return(a) );
phi(n) := block( [a:n,f:0], if n=1 then return(0), for f:1 thru n do if gcd(f,n)>1 then a : a - 1, return(a) );
Nc(n) := block( [a:0,d:1], if n=1 then return(1), for d in divisors(n) do if n>d then a : a + Nc(d) * phi(n/d), return(a) );
Nd(n) := block( [a:0,d:1], for d in divisors(n) do if n>d then a : a + Na(d) * phi(n/d), return(a) );
Nm(n) := Na(n) - Nd(n);
List31: makelist(i,i,1,31);
map(Na, List31);
map(phi, List31);
map(Nc, List31);
map(Nd, List31);
map(Nm, List31);

See also Exploring, largest mu-atoms, largest islands, Phi(N), and this page that lists various integer sequences.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index

This work is licensed under a Creative Commons Attribution 2.5 License. Details here s.13