Periodic points of complex quadratic mappings

Definitions

Let

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

be the complex quadratic mapping, where z and c are complex numbers.

Notationally, f^{(k)} _c (z) is the k-fold composition of f_c with itself (not to be confused with the kth derivative of f_c)—that is, the value after the k-th iteration of the function f _c. Thus

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

Periodic points of a complex quadratic mapping of period p are points z of the dynamical plane such that

:f^{(p)} _c (z) = z,

where p is the smallest positive integer for which the equation holds at that z.

We can introduce a new function:

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

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

:F_p(z,f) = 0,

which is a polynomial of degree 2^p.

Number of periodic points

The degree of the polynomial F_p(z,f) describing periodic points is d = 2^p so it has exactly d = 2^p complex roots (= periodic points), counted with multiplicity.

Stability of periodic points (orbit) - multiplier

The multiplier (or eigenvalue, derivative) m(f^p,z_0)=\lambda of a rational map f iterated p times at cyclic point z_0 is defined as:

:m(f^p,z_0) = \lambda = \begin{cases} f^{p \prime}(z_0), &\mbox{if }z_0 \ne \infty \ \frac{1}{f^{p \prime} (z_0)}, & \mbox{if }z_0 = \infty \end{cases}

where f^{p\prime} (z_0) is the first derivative of f^p with respect to z at z_0.

Because the multiplier is the same at all periodic points on a given orbit, it is called a multiplier of the periodic orbit.

The multiplier is:

• a complex number;
• invariant under conjugation of any rational map at its fixed point;
• used to check stability of periodic (also fixed) points with stability index abs(\lambda). ,

A periodic point is

• attracting when abs(\lambda)
  • super-attracting when abs(\lambda) = 0;
  • attracting but not super-attracting when 0
• indifferent when abs(\lambda) = 1;
  • rationally indifferent or parabolic if \lambda is a root of unity;
  • irrationally indifferent if abs(\lambda)=1 but multiplier is not a root of unity;
• repelling when abs(\lambda) 1.

Periodic points

• that are attracting are always in the Fatou set;
• that are repelling are in the Julia set;
• that are indifferent fixed points may be in one or the other. A parabolic periodic point is in the Julia set.

Period-1 points (fixed points)

Finite fixed points

Let us begin by finding all finite points left unchanged by one 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 as

: \ z^2-z+c=0.

Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:

: \alpha_1 = \frac{1-\sqrt{1-4c}}{2} and \alpha_2 = \frac{1+\sqrt{1-4c}}{2}. So for c \in \mathbb{C} \setminus {1/4} 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},

we have \alpha_1 + \alpha_2 = 1.

Thus fixed points are symmetrical about z = 1/2.

Complex dynamics

Here different notation is commonly used:

:\alpha_c = \frac{1-\sqrt{1-4c}}{2} with multiplier \lambda_{\alpha_c} = 1-\sqrt{1-4c}

and

:\beta_c = \frac{1+\sqrt{1-4c}}{2} with multiplier \lambda_{\beta_c} = 1+\sqrt{1-4c}.

Again we have

:\alpha_c + \beta_c = 1 .

Since the derivative with respect to z is

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

we have

:P_c'(\alpha_c) + P_c'(\beta_c)= 2 \alpha_c + 2 \beta_c = 2 (\alpha_c + \beta_c) = 2 .

This implies that P_c can have at most one attractive fixed point.

These points are distinguished by the facts that:

• \beta_c is:
  • the landing point of the external ray for angle=0 for c \in M \setminus \left{ 1/4 \right}
  • the most repelling fixed point of the 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).
• \alpha_c is:
  • the landing point of several rays
  • attracting when c is in the main cardioid of the Mandelbrot set, in which case it is in the interior of a filled-in Julia set, and therefore belongs to the Fatou set (strictly to the basin of attraction of finite fixed point)
  • parabolic at the root point of the limb of the Mandelbrot set
  • repelling for other values of c

Special cases

An important case of the quadratic mapping is c=0. In this case, we get \alpha_1 = 0 and \alpha_2=1. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

Only one fixed point

We have \alpha_1=\alpha_2 exactly when 1-4c=0. This equation has one solution, c=1/4, in which case \alpha_1=\alpha_2=1/2. In fact c=1/4 is the largest positive, purely real value for which a finite attractor exists.

Infinite fixed point

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

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

and extend f_c such that f_c(\infty)=\infty.

Then infinity is:

• superattracting
• a fixed point of f_c:f_c(\infty)=\infty=f^{-1}_c(\infty).

Period-2 cycles

Period-2 cycles are two distinct points \beta_1 and \beta_2 such that f_c(\beta_1) = \beta_2 and f_c(\beta_2) = \beta_1, and hence

:f_c(f_c(\beta_n)) = \beta_n

for n \in {1, 2}:

:f_c(f_c(z)) = (z^2+c)^2+c = z^4 + 2cz^2 + c^2 + c.

Equating this to z, we obtain

:z^4 + 2cz^2 - z + c^2 + c = 0.

This equation is a polynomial of degree 4, and so has four (possibly non-distinct) solutions. However, we already know two of the solutions. They are \alpha_1 and \alpha_2, computed above, since if these points are left unchanged by one application of f, then clearly they will be unchanged by more than one application of f.

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

First method of factorization

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

This expands directly as x^4 - Ax^3 + Bx^2 - 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 two solutions, and only need the other two. Hence the problem is equivalent to 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 = c \beta_1 \beta_2 and A = 1 + \beta_1 + \beta_2. Matching these against the coefficients from expanding f, we get

: D = c \beta_1 \beta_2 = c^2 + c and A = 1 + \beta_1 + \beta_2 = 0.

From this, we easily get

:\beta_1 \beta_2 = c + 1 and \beta_1 + \beta_2 = -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 that:

:f_c(\beta_1) = \beta_2 and f_c(\beta_2) = \beta_1,

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

Second method of factorization

We can factor the quartic by using polynomial long division to divide out the factors (z-\alpha_1) and (z-\alpha_2), which account for the two fixed points \alpha_1 and \alpha_2 (whose values were given earlier and which still remain at the fixed point after two iterations):

:(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. They are repelling outside the main cardioid.

The second factor has the two roots

:\frac{-1 \pm \sqrt{-3 -4c}}{2}. ,

These two roots, which are the same as those found by the first method, form the period-2 orbit.

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. We have | \beta_1 | = | \beta_2 | = 1. Thus, both these points are "hiding" in the Julia set. Another special case is c=-1, which gives \beta_1 = 0 and \beta_2 = -1. This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

Cycles for period greater than 2

The degree of the equation f^{(n)}(z)=z is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8. After factoring out the factors giving the two fixed points, we would have a sixth degree equation.

There is no general solution in radicals to polynomial equations of degree five or higher, so the points on a cycle of period greater than 2 must in general be computed using numerical methods. However, in the specific case of period 4 the cyclical points have lengthy expressions in radicals.

In the case c = –2, trigonometric solutions exist for the periodic points of all periods. The case z_{n+1}=z_n^2-2 is equivalent to the logistic map case r = 4: x_{n+1}=4x_n(1-x_n). Here the equivalence is given by z=2-4x. One of the k-cycles of the logistic variable x (all of which cycles are repelling) is

:\sin^2\left(\frac{2\pi}{2^k-1}\right), , \sin^2\left(2\cdot\frac{2\pi}{2^k-1}\right), , \sin^2\left(2^2\cdot\frac{2\pi}{2^k-1}\right), , \sin^2\left(2^3\cdot\frac{2\pi}{2^k-1}\right), \dots , \sin^2\left(2^{k-1}\frac{2\pi}{2^k-1}\right).

References

References

1. Alan F. Beardon, Iteration of Rational Functions, Springer 1991, {{ISBN. 0-387-95151-2, p. 41
2. Alan F. Beardon, ''Iteration of Rational Functions'', Springer 1991, {{ISBN. 0-387-95151-2, page 99
3. "Some Julia sets by Michael Becker".
4. [http://www.math.nagoya-u.ac.jp/~kawahira/works/cauliflower.pdf On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178. ] {{webarchive. link. (2011-07-17)
5. [http://www.ibiblio.org/e-notes/MSet/Attractor.htm Periodic attractor by Evgeny Demidov] {{webarchive. link. (2008-05-11)
6. [[Robert L. Devaney. R L Devaney]], [[Linda Keen. L Keen]] (Editor): Chaos and Fractals: The Mathematics Behind the Computer Graphics. Publisher: Amer Mathematical Society July 1989, {{ISBN. 0-8218-0137-6 , {{ISBN. 978-0-8218-0137-6
7. [http://www.ibiblio.org/e-notes/MSet/Attractor.htm Period 2 orbit by Evgeny Demidov] {{webarchive. link. (2008-05-11)
8. [https://arxiv.org/abs/0802.2565 Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram]