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

**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

Figure 8.1. The Julia sets ** J(c)**
for

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

Figure 8.2. The Julia sets ** J(c)**
for

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

Figure 8.3. Julia sets for ** c = -1.77289
**and

Figure 8.4. Julia sets for ** c = -1.75778+0.0137961
I **and

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

Figure 8.5. The Julia set ** J(c0)**
near

Figure 8.6. The Julia set ** J(c0)**
near

Figure 8.7. The Julia set ** J(c0)**
near

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

On the other hand experiments show that if we zoom in to a preperiodic
point in ** M**, the nearby miniature

The self-similarity ratios ** m** of
Theorem 7.8 for the three values of

In Figure 8.7 we are looking
at successive magnification by a factor of 3. ** |m^4| = 3.11329**,
and

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

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

after a finite number of iterationsof the function ** g. **Let

`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

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

where ** O[e^2] **denotes a quantity no larger than a fixed
multiple of

`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

To account for the similarity between ** M** near

`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

We need to compute the numbers ** k[n]**.Using the identities

** 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])
+ ... `

This can be thought of as the ** n**th partial sum in an
infinite series, and we need to show that it converges. If it does, then
its sum is the consatnt

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

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

for some constants ** A** and

`K = A + 2h*B/(2h-1)`

provided that ** |2h| > 1**, and we have already seen
that this holds whenver

*This page was last revised on February 10, 1998.*