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.

)—that is, the value after the k-th iteration of the function

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

: points z satisfying which is a polynomial of degree

complex roots (= periodic points), counted with multiplicity.

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

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

Here different notation is commonly used:[4] and Again we have Since the derivative with respect to z is we have This implies that

These points are distinguished by the facts that: An important case of the quadratic mapping is

In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.

is the largest positive, purely real value for which a finite attractor exists.

to the Riemann sphere (extended complex plane)

Then infinity is: Period-2 cycles are two distinct points

, computed above, since if these points are left unchanged by one application of

Our 4th-order polynomial can therefore be factored in 2 ways: This expands directly as

(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.

, we get From this, we easily get From here, we construct a quadratic equation with

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.

(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.

Thus, both these points are "hiding" in the Julia set.

This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

is 2n; thus for example, to find the points on a 3-cycle we would need to solve an equation of degree 8.

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.

is equivalent to the logistic map case r = 4:

One of the k-cycles of the logistic variable x (all of which cycles are repelling) is