In these notes I will use mathematical notation similar to the syntax of Mathematica.

6. Relations and symmetries, February 4.

Relation between the logistic and classical Mandelbrot sets.

We have considered the logistic and classical Mandlebrot sets, L and M, shown below.

Figure 6.1. The logistic and classical Mandelbrot sets L and M.

Figure 6.2. Close-ups of L (at r = 2.7 + 1.2I) and M (at c = -.1 + I) near period 3 circles, each magnified 5 times.

How are they related? they have large scale differences but small scale similarities. Each is related to a family of quadratic functions and indicates which member of the family have bounded critical orbits. For L the function is

f[x_] := r*x*(1-x)                            (6.3)

and for M it is

g[z_] := z^2 + c.                             (6.4)

If we define

z[r_, x_] := r*(1/2 - x),                         (6.5)

then we find that

z[r, f[x]] = z[r, x]^2 + r/2 - r^2/4.

Hence if we set

c[r_] := r(2 - r)/4,                                                            (6.6)

then we have

z[r, f[x]] = g[z[r, x]].

This means that (6.5) and (6.6) give us a way to transform the orbits of f (6.3) to those of g (6.4). Note that

z[r, 1/2] = 0,

so the critical point of f, x = 1/2, gets transformed to the critical point of g, z = 0. Thus the critical orbit of f for a given value of r transforms to the critical orbit of g for c = r/2 - r^2/4, (6.6) tells us how to map L to M. The following table shows some interesting values of c[r].

 r 1 u 2-u 1 + Sqrt[5-u] 1 - Sqrt[5-u] c[r] 1/4 u/2-u^2/4 u/2-u^2/4 1 - u/4 1 - u/4

This shows that

• The point where the two large circles meet (r = 1) in L maps to the cusp point (c = 1/4) of M.
• The two largest circles (which make up the main body of L), each map to the main body of M, the cardioid.
• The second two largest circles (which make up the period 2 body of L), each map to the period 2 body of M, the largest circle.

Moreover, for every value of c other than 1/4, there are two corresponding values of r, namely 1 + Sqrt[1-4c] and 1 - Sqrt[1-4c].

Symmetry in the two Mandelbrot sets.

Both the logistic and classical Mandelbrot sets appear to be symmetric about the horizontal axis, i.e., the bottom half of each is a mirror image of the top half.

This has to do with conjugation of the complex numbers involved. Recall that if u = u1 + u2*I is a complex number (with u1 and u2 being real numbers), then the complex conjugate of u is

Conj[u] = u1 - u2*I

One sees easily that for any complex numbers a and b, conjugation obeys the following rules.

Conj[a + b] = Conj[a] + Conj[b]

Conj[a*b] = Conj[a]*Conj[b]

|Conj[a]| = |a|

Conj[Conj[a]] = a.

The observed symmetries in L and M can be expressed in terms of conjugation as follows:

• A complex number r is in L if and only if its conjugate Conj[r] is in L.
• A complex number c is in M  if and only if its conjugate Conj[c] is in M.

Why is this true? By the definitions of L and M, these staemmnts are equivalent to the following:

• The critical orbit for the function f[x] for r is bounded if and only if the critical orbit ofthe function f[x] for Conj[r] is bounded.
• The critical orbit for the function g[z] for c is bounded if and only if the critical orbit ofthe function g[z] for Conj[c] is bounded.

Using the rules of conjugation listed above, one can show that each number in the critical orbit of f for Conj[r] is the conjugate of the corresponding number in the critical orbit of f for r. Since conjugation does not change absolute value, it does not change the boundedness of an orbit. Hence the critical orbit and its conjugate are either both bounded or both unbounded.

The same can be said about critical orbits of g. This accounts for the symmetry about the horizontal axis of both L and M.

The logistic Mandelbrot set L is also appears to be symmetric about the center of the picture, i.e., it is unchanged by rotation about 180 degrees. Since the center of the picture is at r = 1, this symmetry can be expressed as follows.

• A complex number r is in L if and only if 2 - r is in L.

To see that this is true, note that by (6.6) both r and 2 - r map to the same value of c, so they are either both in L or both not in L. This accounts for the rotational symmetry in L.

Symmetry in Julia sets.

Recall that the classical Julia set J(c) is the set of numbers z having bounded orbits under the function g[z] of (6.4). We define the logistic Julia set LJ(r) to be the set of numbers z having bounded orbits under the function f[x] of (6.3). Both can be obtained with Fractint by right-clicking on the corrsponding Mandelbrot image.

There are several symmetry issues to discuss here.

1. Julia sets, both logistic and classical, appear to have rotational symmetry.
2. The classical Julia set for Conj[c] is the mirror image of the one for c, and similarly for logistic Julia sets.
3. There is a close relation between LJ(r), LJ(2-r) and J(r*(2-r)/4).

The first of these can be reformulated as

A complex number z is has a bounded orbit if and only if -z has a bounded orbit.

To see, this, note that g[z] = g[-z], so the orbits of z and -z coincide beyond the first point. Thus either both are bounded or both are unbounded. In the logistic case we have f[x] = f[1-x], which leads to rotational symmetry around the point x = 1/2.

The second can be reformulated as

A complex number z is in the Julia set J(c) if and only if Conj[z] in in J(Conj[c]).

For this note the orbit of Conj[z] under the function z^2 + Conj[c] is conjugate to the orbit of z under the function z^2 + c. Thus either both are bounded or both are unbounded. The same can be said about logistic Julia sets.

We will illustrate the third type of symmetry by looking at it in the case r = 2.56 + I.

Figure 6.7. The logistic Julia sets LJ(r) and LJ(2-r) for r = 2.56 + I. The region shown in both pictures is               x = x1 + x2*I for -.833 < x1 < 1.833 and -1 < x2 < 1.

Figure 6.8. The classical Julia set J(r*(2-r)/r) for r = 2.56 + I. The region shown is z = z1 + z2*I for      -2 < z1 < 2 and -1.5 < z2 < 1.5.

The three Julia sets shown in Figures 6.7 and 6.8 are all shown at the same scale. They appear to be the same up to change of size and orientation. The centers of the two pictures in 6.7 are at x = 1/2, and the one in 6.8 is at z = 0.

According to (6.5), a point 1/2 + y in the left picture of 6.7 maps to the point -r*y in 6.8. Since |r| = 2.74838, (the Mathematica syntax for this is Abs[r]), the Julia set shown in 6.8 is 2.74838 times the size of the one in the left picture of 6.7. We also have

|2-r| = 1.14612 and |r/(2-r)| = 2.39798,

the image in 6.8 is 1.14612 times the size of the right image in 6.7, and the latter is 2.39798 times the size of the left image. The difference in the orientations of the three pictures is related to the fact that multiplication by a complex number s = s1 + s2*I induces counterclockwise rotation through the angle ArcTan[s2/s1].