Periodic points of complex quadratic mappings

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This article describes periodic points of some complex quadratic map. This theory is applied in relation with the theories of Fatou and Julia sets.

Contents

[edit] Definitions

Let

f_c(z)=z^2+c\,

where z and c are complex-valued. (This \ f is the complex quadratic mapping mentioned in the title.) This article explores the periodic points of this mapping - that is, the points that form a periodic cycle when \ f is repeatedly applied to them.

\ f^{(k)} _c (z) is the \ k -fold compositions of f _c\, with itself = iteration of function f _c\,

\ f^{(k)} _c (z) = f_c(f^{(k-1)} _c (z))

then periodic points of complex quadratic mapping of period \ p are points \ z of dynamical plane such that :

\ z : f^{(p)} _c (z) = z

where \ p is the smallest positive integer.

We can introduce new function:

\ F_p(z,f) = f^{(p)} _c (z) - z

so periodic points are zeros of function \ F_p(z,f)  :

\ z : F_p(z,f) = 0

which is polynomial of degree \ = 2^p

[edit] Stability of periodic points

The multiplier m(f,z_0)=\lambda \, of fixed point z_0\, is defined as

m(f,z_0)=\lambda = \begin{cases} f_c'(z_0), &\mbox{if }z_0\ne \infty \\ \frac{1}{f_c'(z_0)}, & \mbox{if }z_0 = \infty \end{cases}

where f_c'(z_0)\, is first derivative of \ f_c with respect to z\, at z_0\,.

Multiplier is:

  • complex number,
  • invariant under conjugation of any rational map at its fixed point[1]
  • used to check stability of periodic (also fixed) points.

[edit] Period-1 points (fixed points)

[edit] Finite fixed points

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Let us begin by finding all finite points left unchanged by 1 application of f. These are the points that satisfy \ f_c(z)=z. That is, we wish to solve

z^2+c=z\,

which can be rewritten

\ z^2-z+c=0.

Since this is an ordinary quadratic equation in 1 unknown, we can apply the standard quadratic solution formula. Look in any standard mathematics textbook, and you will find that there are two solutions of \ Ax^2+Bx+C=0 are given by

x=\frac{-B\pm\sqrt{B^2-4AC}}{2A}

In our case, we have A = 1,B = − 1,C = c, so we will write

\alpha_1 = \frac{1-\sqrt{1-4c}}{2} and \alpha_2 = \frac{1+\sqrt{1-4c}}{2}.

So for c \in C \setminus [1/4,+\inf ] we have two finite fixed points \alpha_1 \, and \alpha_2\, .

Since

\alpha_1 = \frac{1}{2}-m and \alpha_2 = \frac{1}{2}+ m where m = \frac{\sqrt{1-4c}}{2}

then \alpha_1 + \alpha_2 = 1 \,.

It means that fixed points are symmetrical around z = 1/2\,.

This image shows fixed points for c=i
This image shows fixed points for c=i

[edit] Complex dynamics

Here different notation is commonly used:

\alpha_c = \frac{1-\sqrt{1-4c}}{2} and \beta_c = \frac{1+\sqrt{1-4c}}{2}.

Using Viète's formulas one can show that:

\alpha_c + \beta_c = -\frac{B}{A} = 1

Since derivative with respect to z is :

P_c'(z) = \frac{d}{dz}P_c(z) = 2z

then

P_c'(\alpha_c) + P_c'(\beta_c)= 2 \alpha_c + 2 \beta_c = 2 (\alpha_c + \beta_c) = 2 \,

It implies that P_c \, can have at most one attractive fixed point.

This points are distinguished by the facts that:

  • \beta_c \, is :
    • the landing point of external ray for angle=0 for c \in M \setminus \left \{ \frac{1}{4} \right \}
    • the most repelling fixed point, belongs to Julia set,
    • the one on the right ( whenever fixed point are not symmetrical around the real axis), it is the extreme right point for connected Julia sets (except for cauliflower)[2].
  • \alpha_c \, is:
    • landing point of several rays
    • is :
      • attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, it means belongs to Fatou set ( strictly to basin of attraction of finite fixed point )
      • parabolic at the root point of the limb of Mandelbtot set
      • repelling for other c values
Fatou set for F(z)=z*z with marked fixed point
Fatou set for F(z)=z*z with marked fixed point

[edit] Special cases

An important case of the quadratic mapping is c = 0. In this case, we get α1 = 0 and α2 = 1. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

[edit] Only one fixed point

We might wonder what value c should have to cause α1 = α2. The answer is that this will happen exactly when 1 − 4c = 0. This equation has 1 solution: c = 1 / 4 (in which case, α1 = α2 = 1 / 2). This is interesting, since c = 1 / 4 is the largest positive, purely-real value for which a finite attractor exists.

[edit] Infinite fixed point

We can extend complex plane \mathbb{C} to Riemann sphere (extended complex plane) \mathbb{\hat{C}} by adding infinity

\mathbb{\hat{C}} = \mathbb{C} \cup \{ \infty \}

and extend polynomial f_c\, such that f_c(\infty)=\infty\,

Then infinity is :

  • superattracting
  • fixed point f_c(\infty)=\infty=f^{-1}_c(\infty)\,

of polynomial f_c\,[3].

[edit] Period-2 cycles

Suppose next that we wish to look at period-2 cycles. That is, we want to find two points β1 and β2 such that fc(β1) = β2, and fc(β2) = β1.

Let us start by writing fc(fc(βn)) = βn, and see where trying to solve this leads.

f_c(f_c(z)) = (z^2+c)^2+c = z^4 + 2z^2c + c^2 + c.\,

Thus, the equation we wish to solve is actually z4 + 2cz2 − z + c2 + c = 0.

This equation is a polynomial of degree 4, and so has 4 (possibly non-distinct) solutions. However, actually, we already know 2 of the solutions. They are α1 and α2, computed above. It is simple to see why this is; if these points are left unchanged by 1 application of f, then clearly they will be unchanged by 2 applications (or more).

Our 4th-order polynomial can therefore be factored in 2 ways :

[edit] first method

(z-\alpha_1)(z-\alpha_2)(z-\beta_1)(z-\beta_2) = 0.\,

This expands directly as x4 − Ax3 + Bx2 − Cx + D = 0 (note the alternating signs), where

D = \alpha_1 \alpha_2 \beta_1 \beta_2\,
C = \alpha_1 \alpha_2 \beta_1 + \alpha_1 \alpha_2 \beta_2 + \alpha_1 \beta_1 \beta_2 + \alpha_2 \beta_1 \beta_2\,
B = \alpha_1 \alpha_2 + \alpha_1 \beta_1 + \alpha_1 \beta_2 + \alpha_2 \beta_1 + \alpha_2 \beta_2 + \beta_1 \beta_2\,
A = \alpha_1 + \alpha_2 + \beta_1 + \beta_2.\,

We already have 2 solutions, and only need the other 2. This is as difficult as solving a quadratic polynomial. In particular, note that

\alpha_1 + \alpha_2 = \frac{1-\sqrt{1-4c}}{2} + \frac{1+\sqrt{1-4c}}{2} = \frac{1+1}{2} = 1

and

\alpha_1 \alpha_2 = \frac{(1-\sqrt{1-4c})(1+\sqrt{1-4c})}{4} = \frac{1^2 - (\sqrt{1-4c})^2}{4}= \frac{1 - 1 + 4c}{4} = \frac{4c}{4} = c.

Adding these to the above, we get D = cβ1β2 and A = 1 + β1 + β2. Matching these against the coefficients from expanding f, we get

D = cβ1β2 = c2 + c and A = 1 + β1 + β2 = 0.

From this, we easily get : β1β2 = c + 1 and β1 + β2 = − 1.

From here, we construct a quadratic equation with A' = 1,B = 1,C = c + 1 and apply the standard solution formula to get

\beta_1 = \frac{-1 - \sqrt{-3 -4c}}{2} and \beta_2 = \frac{-1 + \sqrt{-3 -4c}}{2}.

Closer examination shows (the formulas are a tad messy) that :

fc(β1) = β2 and fc(β2) = β1

meaning these two points are the two halves of a single period-2 cycle.

[edit] Second method of factorization[4]

(z^2+c)^2 + c -z = (z^2 + c - z)(z^2 + z + c +1 ) \,

The roots of the first factor are the two fixed points z_{1,2}\, . They are repelling outside the main cardioid.

The second factor has two roots

z_{3,4} = -\frac{1}{2} \pm (-\frac{3}{4} - c)^\frac{1}{2} \,

These two roots form period-2 orbit.

[edit] Special cases

Again, let us look at c = 0. Then

\beta_1 = \frac{-1 - i\sqrt{3}}{2} and \beta_2 = \frac{-1 + i\sqrt{3}}{2}

both of which are complex numbers. By doing a little algebra, we find | β1 | = | β2 | = 1. Thus, both these points are "hiding" in the Julia set.

Another special case is c = − 1, which gives β1 = 0 and β2 = − 1. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

[edit] Cycles for period>2

There is no general solution in radicals to polynomial equations of degree five or higher, so it must be computed using numerical methods.

[edit] References

  1. ^ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
  2. ^ Periodic attractor by Evgeny Demidov
  3. ^ R L Devaney, L Keen (Editor}: Chaos and Fractals: The Mathematics Behind the Computer Graphics. Publisher: Amer Mathematical Society July 1989, ISBN-10: 0821801376 , ISBN-13: 9780821801376
  4. ^ Period 2 orbit by Evgeny Demidov

[edit] Further reading

[edit] External links