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The Structure of the Mandelbrot Set

The Mandelbrot set is surely one of the most visually fascinating structures in mathematics. The Mandelbrot set is the set of complex numbers c such that the iteration scheme

zn+1 = zn² + c

is bounded when starting from the point z0=0. A significant subset of the Mandelbrot set consists of those values of c such that the iteration scheme approaches limits for which

zn+1 = zn.

Such a limit point z* satisfies the equation

z* = z*² +c

For any c there is a limit point z*; i.e., such that if z0=z* the iteration will remain at z* forever.

The crucial question is what are the limit points that are stable so that the iteration starting from z0=0 will approach them.

Consider the deviations of the iteration values from the corresponding limit point; i.e.,

zn+1 = zn² + c
z* = z*² + c
Subtraction gives
zn+1-z* = zn² - z*² = (zn-z*)(zn+z*)


|zn+1-z*| will be less than |zn-z*| if |zn+z*|<1

For values of zn close to z* this reduces to 2|z*|<1. The boundary between the stable and unstable limit points is given by |z*|=1/2. Such limit points are given by the equation

z* = ½e
for 0≤φ≤2π

The question is what are the values of c which give those limit points. Those values of c are simply

c = z* - z*² = = ½e(1-½e)
c = ½cos(φ)−¼cos(2φ) + i[½sin(φ)−¼sin(2φ)]

This equation is a parametric equation for the set of c values. It shows how the points on the circle of radius 1/2 in the z* space map into the c space. For example, z*=½ maps into c=¼, z*=−½ maps into c=−3/4, z*=i½ maps into c=½+i¼ and z*=−i½ maps into c=−½+i¼.

The plot below shows the full set of c values.

And there is the familiar cardioid shape that bounds the main body of the mandelbrot set, as seen below.

The Limiting 2-Cycles

The iteration may approach a limit cycle rather than a limit point. For a two-period cycle of z1* and z2* the values would have to satisfy the equations

z2* = z1*² + c
z1* = z2*² + c

From these equations defining z1* and z2* it follows that

z2*−z1* = -(z2*−z1*)(z2*+z1*)
and with division by (z2*−z1*)
which is valid if z2*≠z1*
1 = −(z2*+z1*)
and therefore
z2* = −1 − z1* = −(1 + z1*)

The deviations from the limit values satisfy the equations

zn+2-z2* = (zn+1-z1*)(zn+1+z1*)
zn+1-z1* = (zn-z2*)(zn+z2*)
zn+2-z2* = (zn-z2*)(zn+z2*)(zn+1+z1*)

Therefore if |zn+2-z2*| is to be less than |zn-z2*| it must be that

|(zn+z2*)(zn+1+z1*)| < 1

For values very close to a cycle pair this reduces to:

|(2z2*)(2z1*)| < 1
or, equivalently
|z2*z1*| < ¼

For the boundary of the stable set equality prevails; i.e.,

|z2*z1*| = ¼

It was previously determined that z2* = −(1 + z1*), therefore the condition for the boundary of stability reduces to:

|(1+z1*)z1*| = ¼

The allowable values of z1 are solutions to the quadratic equation

z(1+z) = ¼e
or, equivalently
z² + z - ¼e = 0

There are two solutions to this quadratic equation; one corresponding to z1* and the other to z2*.

It is not the solutions for z1* or z2* that are of most interest; it is the set of values of c corresponding to these solutions.

The defining equation

z2* = z1*² + c
combined with the result that
z2* = −(1 + z1*)
−(1 + z1*) = z1*² + c.
c = −(1 + z1* + z1*²)

It is known that

z1* + z1*² = ¼e
c = −(1 + ¼e)

This is the locus of a circle of radius ¼, the center of which is shifted one unit to the left on the real axis. This is shown below in a diagram with the values of c corresponding to limit points.

The Limiting 3-Cycles

The analysis for 3-cycles is more difficult but by same procedures as in the previous it can be established that the boundaries between the stable and unstable cycles are circles of radius 1/8. In general the boundaries between stable and unstable m-cycles are circles of radius 1/2m. The diagram below shows the placement of circles having radii of 1/8, 1/16 and 1/32.

There are also easily discernible circles in the Mandelbrot set of radius 1/64.

The limit points z1*, z2* and z3* satisfy the equations

z3* = z2*² + c
z2* = z1*² + c
z1* = z3*² + c

The substitution of the equation for z1* into the equation for z2*² and the substitution of that equation into the equation for z3* produces a polynomial equation of degree 8.

z*3 = ((z*3²+c)² + c)² + c
= ((z*34+2z*3²c + c² + c)² + c
= (z*34+2z*3²c)² + 2(z*34+2z*3²c)(c² + c) + (c² + c)² + c

There will be eight solutions to this equation and two pairs of triples which satisfy the conditions for limiting 3-cycles for each given value of c. The product of the eight solutions will be equal to the constant term divided by the coefficient of the highest power of z*. This means that the product of the solutions is equal to (c² + c)² + c.

In order to establish the conditions for stability of the 3-cycles consider the deviations between zn+3 and z3*, zn+2 and z2*, zn+1 and z1*, and zn and z3*. These deviations satisfy the equations

zn+3 − z3* = (zn+2 − z2*)(zn+2 + z2*)
zn+2 − z2* = (zn+1 − z1*)(zn+1 + z1*)
zn+1 − z1* = (zn − z3*)(zn + z3*)

Successive substitutions establish that if |zn+3 − z3*| is to be less than |zn − z3*| it is necessary that

|(zn+2 + z2*)(zn+1 + z1*)(zn + z*)| < 1

For values very close to the limiting 3-cycles this reduces to

|(2z2*)(2z1*)(2z3*)| < 1
or, equivalently
2³|z1*z2*z3*| < 1

and for the boundary between the stable and unstable 3-cycles

|z1*z2*z3*| = 1/8

From the defining equations for the limit points it follows that

z3* − z2* = (z2* − z1*)(z2* + z1*)
z2* − z1* = (z1* − z3*)(z1* + z3*)
z3* − z1* = (z2* − z3*)(z2* + z3*)

and therefore

z3* − z1* = (z1* − z3*)(z3* + z2*)(z2* + z1*)(z1* + z3*)
and with division by z3* − z1*
1 = −(z3* + z2*)(z2* + z1*)(z1* + z3*)

(To be continued.)

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