Periodic points of complex quadratic mappings

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.
These periodic points play a role in the theories of Fatou and Julia sets.
Definitions.
Let 
be the complex quadric mapping, where formula_2 and formula_3 are complex-valued. 
Notationally, formula_4 is the formula_5 -fold composition of formula_6 with itself—that is, the value after the "k"-th iteration of function formula_7 Thus
Periodic points of a complex quadratic mapping of periodformula_9 are pointsformula_10 of the dynamical plane such that 
whereformula_9 is the smallest positive integer for which the equation holds at that "z".
We can introduce a new function: 
so periodic points are zeros of function formula_14: points "z" satisfying
which is a polynomial of degree formula_16
Number of periodic points.
Degree of polynomial formula_14 describing periodic points is formula_18 so it has exactly formula_18 complex roots (= periodic points), counted with multiplicity,
Stability of periodic points (orbit) - multiplier.
 
The multiplier (or eigenvalue, derivative) formula_20 of a rational map formula_21 iterated formula_22 times at cyclic point formula_23is defined as:
where formula_25 is the first derivative offormula_26 with respect to formula_27 at formula_28.
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 periodic point is
Periodic points
Period-1 points (fixed points).
Finite fixed points.
Let us begin by finding all points left unchanged by one application of formula_37. These are the points that satisfy formula_38. That is, we wish to solve
which can be rewritten as
Since this is an ordinary quadratic equation in one unknown, we can apply the standard quadratic solution formula:
So for formula_43 we have two fixed points formula_44 and formula_45.
Since 
then formula_49.
Thus fixed points are symmetrical around formula_50.
Complex dynamics.
Here different notation is commonly used:
and 
Using Viète's formulas one can show that:
Since the derivative with respect to z is 
then 
This implies that formula_58 can have at most one attractive fixed point. 
These points are distinguished by the facts that:
Special cases.
An important case of the quadratic mapping is formula_64. In this case, we get formula_65 and formula_66. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.
Only one fixed point.
We have formula_67 exactly when formula_68 This equation has one solution, formula_69 in which case formula_70. In fact formula_71 is the largest positive, purely real value for which a finite attractor exists.
Infinite fixed point.
We can extend the complex plane formula_72 to the Riemann sphere (extended complex plane) formula_73 by adding infinity :
and extending polynomial formula_75such that formula_76
Then infinity is :
Period-2 cycles.
Period-2 cycles are two distinct points formula_79 and formula_80 such that formula_81 and formula_82.
We write formula_83
Equating this to "z", we obtain
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 formula_86 and formula_87, computed above, since if these points are left unchanged by one application of formula_37, then clearly they will be unchanged by more than one application of formula_37.
Our 4th-order polynomial can therefore be factored in 2 ways:
First method of factorization.
This expands directly as formula_91 (note the alternating signs), where
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
and
Adding these to the above, we get formula_98 and formula_99. Matching these against the coefficients from expanding formula_37, we get
From this, we easily get
From here, we construct a quadratic equation with formula_105 and apply the standard solution formula to get
Closer examination shows that :
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 formula_110 and formula_111 which account for the two fixed points formula_86 and formula_87 (whose values were given earlier and which still remain at the fixed point after two iterations):
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
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 formula_64. Then
both of which are complex numbers. We have formula_119. Thus, both these points are "hiding" in the Julia set.
Another special case is formula_120, which gives formula_121 and formula_122. 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 formula_123 is 2"n"; 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 formula_124 is equivalent to the logistic map case "r" = 4: formula_125 Here the equivalence is given by formula_126 One of the "k"-cycles of the logistic variable "x" (all of which cycles are repelling) is 