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

8. Julia sets for periodic and preperiodic points.

Recall that the Julia set J(c) is the set of z having bounded orbits under the function g[z] = z^2 + c. We have seen that it is always symmetric with respect to the origin and always contained within the circle |z|=2. Replacing c by its conjugate has the effect of reflection J(c) through the horizontal axis.

Julia sets of periodic points.

The simplest periodic point is c = 0, and in this case one does not need a computer to decide which orbits are bounded. A point z has bounded orbit under the fuinction z^2 if and only if |z| <= 1, so J(0) is the unit disk. Here are some examples of Julia sets for periodic points that are centers of cardioids in miniature Ms, shown in Figures 5.12 and 7.2.

Figure 8.1. The Julia sets J(c) for c = -1.75488 ('the airplane') and c = -0.15652 + 1.03225 I. These points are the centers of cardioids of periods 3 and 4 respectively.

In both cases there one can see several black disks, i.e., replicas of the Julia set J(0). How big are these disks? Are there infinitely many of them? How are they distributed? Where is the critical orbit in each of these pictures? What can you say about orbits of other disks?

Now here are the Julia sets for the periodic points in the two largest buds on the main cardioid of M, at c = -1 (the period 2 bud, whose Julia set is also shown in Figure 5.5) and c = -.122561 + .744862I (the upper period 3 bud shown in Figure 5.13).

Figure 8.2. The Julia sets J(c) for c = -1 and c = -.122561 + .744862I ('Douady's rabbit'). These points are the centers of buds of periods 2 and 3 respectively.

Notice that in J(-1), the black regions appear to meet in pairs, while in Douady's rabbit they meet in in threes.Where is the critical orbit in each of these pictures? What can you say about orbits of other black regions? What happens in the Julia sets for centers of buds of larger periods or secondary buds?

We can also look at the Julia sets for bud centers on miniature M's such as the one with cardioid of period 3 centered at c = -1.75488 shown in Figure 5.12. and the one with the period 4 cardioid centered at c = -0.15652 + 1.03225 I shown in Figure 7.2. If a cardioid has period k, then the replica of a period n bud on it has period k*n. The easiest way to find the centers of such replicated buds is to hunt for them by hand with Fractint and then use FindRoot to get more precision. Using this method we find the following Julia sets contining replicas of J(-1) and Douady's rabbit.

Figure 8.3. Julia sets for c = -1.77289 and c = -0.162868+1.03731 I, showing replicas of J(-1).

Figure 8.4. Julia sets for c = -1.75778+0.0137961 I and c = -0.153941+1.0377 I, showing replicas of Douady's rabbit.

Why do these replications occur? Can we predict the size and orientation of the replica in the center of each picture?

Preperiodic Julia sets and Lei's theorem.

We will now look at the Julia sets associated with the preperiodic points c0 shown in Figures 7.5, 7.6 and 7.7, near the point z = c0. These pictures should be compared with the corresponding areas of the Mandelbrot set.

Figure 8.5. The Julia set J(c0) near z = c0 for c0 = -2, magnified 5, 20 and 80 times. Compare this with Figure 7.5.

Figure 8.6. The Julia set J(c0) near z = c0 for c0 = -0.228155 + 1.1151 I, magnified 5, 15 and 45 times. Compare this with Figure 7.6.

Figure 8.7. The Julia set J(c0) near z = c0 for c0 = -0.101096 + 0.956287 I, magnified 5, 15 and 45 times. Compare this with Figure 7.7.

These pictures differ from the corresponding Mandelbrot pictures in that there are no black regions in them, in contrast to the Julia sets for periodic points. In particular the Julia set J(-2) is just the line segment from -2 to 2. Prove this, i.e., show each z that is not real has an unbounded orbit.

On the other hand experiments show that if we zoom in to a preperiodic point in M, the nearby miniature Ms shrink faster than the view window, so they eventually disappear. Show (either by emprical observation in a few cases or preferably by calculation) that if we shrink the picture by a factor of m as in Theorem 7.7, the miniature Ms shrink by a factor of m^2.

The self-similarity ratios m of Theorem 7.8 for the three values of c0 shown above are 4, 2.83927-1.21255I with |m| = 3.08735, and -0.65517 + 1.15551 I with |m| = 1.32833. The arguments (rotational angles) of the latter two are -23.1256 and 119.553 degrees respectively. This accounts for the slight changes in orientation under successive magnifications in Figure 8.6.

In Figure 8.7 we are looking at successive magnification by a factor of 3. |m^4| = 3.11329, and Arg[m^4] is very close to 120 degrees, which accounts for the 3-fold rotational symmetry in the picture. In the center of the picture one has 3 lines meeting, and there numerous nearby points where 6 lines meet. At each of the letter points there is a tiny replica of M.

The similarity between the Mandelbrot and Julia sets shown above is the subject a theorem of Tan Lei [Lei90]. Here is a partial explanation for it in the case when the period of c0 is 1. (Lei's theorem is good for preperiodic points of any period.) Since c0 is preperiodic, it gets sent to the fixed point

h = (1+ Sqrt[1-4*c0])/2 = m/2

after a finite number of iterationsof the function g. Let g1 denote an iterate of g that does this. Now suppose that z is a point near c0, i.e., z = co + e for a small number e. Then by Taylor's theorem in calculus,

g1[z] = g1[c0] + g1'[c0]*e + R[h,e]*e^2

= h + g1'[c0]*e + R[h,e]*e^2

where R[h,e] is a a function whose absolute value has an upper bound independent of e. The standard mathematical notation for this state of affairs is

g1[z] = h + g1'[c0]*e + O[e^2],

where O[e^2] denotes a quantity no larger than a fixed multiple of e^2. It follows that

g[g1[c0+e]] = (h + g1'[c0]*e + O[e^2])^2 + c0

= h^2 + c0 + 2h*g1'[c0]*e + O[e^2]

= h + m*g1'[c0]*e + O[e^2].

A similar calculation gives the same answer for g[g[g1[c0+e/m]]]. The second order term can be made ignored for purposes of approximation. It follows that the orbit of c0+e is bounded if and only if the orbit a point very close to c0+e/m of is. This accounts for the observed self-similarity in the Julia set J(c0). Give a similar explanation that works for preperiodic c0 of larger period.

To account for the similarity between M near c = c0 and J(c0) near z = z0, we need to show that for small e the critical orbit for c = c0 + e is bounded if and only if the one is for a value of z that is very close to c0 + K*e for a constant K to be named later. This is harder. We need to recall the definition of G given in Lecture 7,

G[0, c_, z_] := z; G[n_, c_, z_] := G[n-1, c, z]^2 + c.

For a given value of c0 we also define

h[n_] := G[n, c0, c0];

d1[n_] := D[G[n, c, c0], c]/. c -> co;

d2[n_] := D[G[n, c0, z], z]/. z -> co;

k[n_] := (d1[n] + d2[n])/d2[n].

Then by Taylor's theorem we have

G[n+1, c0 + e, 0] = G[n, c0 + e, c0 + e]

= G[n, c0, c0 + e] + d1[n]*e + O[e^2]

= G[n, c0, c0] + d2[n]*e + d1[n]*e + O[e^2]

= h[n] + (d1[n] + d2[n])*e + O[e^2]

and

G[n, c0, c0 + e] = h[n] + d2[n]*e + O[e^2]

so

G[n, c0, c0 + k[n]*e] = h[n] + k[n]*d2[n]*e + O[e^2]

= h[n] + (d1[n] + d2[n])*e + O[e^2]

= G[n+1, c0 + e, 0] + O[e^2].

This means that the (n+1)th point in the Mandelbrot orbit of c0 + e can be approximated by the nth point in the Julia orbit of c0 + k[n]*e. Thus we need to show that the numbers k[n] approach a limit K as n goes to infinity. This will mean that the Mandelbrot orbit of c0 + e is bounded if and only if the is Julia orbit of c0 + K*e is bounded, as desired

We need to compute the numbers k[n].Using the identities d1[0] = 0, d2[0] = 1,

d1[n] = 2h[n-1]*d1[n-1] + 1 and d2[n] = 2h[n-1]*d2[n-1],

and defining

H[n_] := Product[h[i], {i, 0, n-1}],

it turns out that

k[n] = Sum[1/(2^i*H[i]),{i, 0, n}]

= 1 + 1/(2*h[0]) + 1/(4*h[0]*h[1]) + ...

+ 1/(2^n*h[0]*...*h[n-1]).                             (8.8)

This can be thought of as the nth partial sum in an infinite series, and we need to show that it converges. If it does, then its sum is the consatnt K that we are looking for.

Now the critical orbit of c0 is

{0, h[0]=c0, h[1] = c0^2 +c0, h[2], ...}.

Since is c0 preperiodic with period 1, this orbit stabilizes to the fixed point h after a finite number of steps. This means that in the series of (8.8), eventually each term is 1/(2h) times the previous term, so we can rewrite it as

A + B(1 + 1/(2h) + 1/(2h)^2 + 1/(2h)^3 + ...)

for some constants A and B. This series converges to

K = A + 2h*B/(2h-1)

provided that |2h| > 1, and we have already seen that this holds whenver c0 is outside the main cardioid. This accounts for the local similaity between the Mandelbrot set near c0 and the Julia set J(c0) near near c0. Give a similar explanation of this local similarity for preperiodic points with larger periods.