# 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 is a complex number. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

This theory is applied in relation with the theories of Fatou and Julia sets.

## Definitions

Let

where and are complex-valued. (This 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 is repeatedly applied to them.

is the -fold compositions of with itself = iteration of function or,

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

where is the smallest positive integer.

We can introduce a new function:

so periodic points are zeros of function :

which is a polynomial of degree

## Stability of periodic points (orbit) - multiplier

The **multiplier** ( or eigenvalue, derivative ) of rational map at fixed point is defined as :

where is first derivative of with respect to at .

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

The multiplier is:

- a complex number,
- invariant under conjugation of any rational map at its fixed point
^{[1]} - used to check stability of periodic (also fixed) points with
**stability index**:

A periodic point is :^{[2]}

- attracting when
- indifferent when
- rationally indifferent or parabolic if is a root of unity
- irrationally indifferent if but multiplier is not a root of unity

- repelling when

Where do periodic points belong?

- attracting is always in Fatou set
- repelling is in the Julia set
- Indifferent fixed points may be in one or the other.
^{[3]}Parabolic periodic point is in Julia set.

## Period-1 points (fixed points)

### Finite fixed points

Let us begin by finding all finite points left unchanged by 1 application of . These are the points that satisfy . That is, we wish to solve

which can be rewritten

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 are given by

In our case, we have , so we will write

So for we have two finite fixed points and .

Since

It means that fixed points are symmetrical around .

#### Complex dynamics

Here different notation is commonly used:^{[4]}

and

Using Viète's formulas one can show that:

Since derivative with respect to z is :

then

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

This points are distinguished by the facts that:

- is :
- the landing point of external ray for angle=0 for
- 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).
^{[5]}

- 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 Mandelbrot set
- repelling for other c values

#### Special cases

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

#### Only one fixed point

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

### Infinite fixed point

We can extend complex plane to Riemann sphere (extended complex plane) by adding infinity

and extend polynomial such that

Then infinity is :

- superattracting
- fixed point of polynomial
^{[6]}

## Period-2 cycles

Suppose next that we wish to look at *period-2 cycles*. That is, we want to find two points and such that , and .

Let us start by writing , and see where trying to solve this leads.

Thus, the equation we wish to solve is actually .

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 and , computed above. It is simple to see why this is; if these points are left unchanged by 1 application of , then clearly they will be unchanged by 2 applications (or more).

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

### First method

This expands directly as (note the alternating signs), where

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

and

Adding these to the above, we get and . Matching these against the coefficients from expanding , we get

From this, we easily get : and .

From here, we construct a quadratic equation with and apply the standard solution formula to get

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

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

### Second method of factorization

The roots of the first factor are the two fixed points . They are repelling outside the main cardioid.

The second factor has two roots

These two roots form period-2 orbit.^{[7]}

#### Special cases

both of which are complex numbers. By doing a little algebra, we find . Thus, both these points are "hiding" in the Julia set. Another special case is , which gives and . This gives the well-known superattractive cycle found in the largest period-2 lobe of the quadratic Mandelbrot set.

## 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.

## References

- ↑ Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2, p. 41
- ↑ Alan F. Beardon,
*Iteration of Rational Functions*, Springer 1991, ISBN 0-387-95151-2, page 99 - ↑ Some Julia sets by Michael Becker
- ↑ On the regular leaf space of the cauliflower by Tomoki Kawahira Source: Kodai Math. J. Volume 26, Number 2 (2003), 167-178.
- ↑ Periodic attractor by Evgeny Demidov
- ↑ R L Devaney, 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
- ↑ Period 2 orbit by Evgeny Demidov

## Further reading

- Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2
- Michael F. Barnsley (Author), Stephen G. Demko (Editor), Chaotic Dynamics and Fractals (Notes and Reports in Mathematics in Science and Engineering Series) Academic Pr (April 1986), ISBN 0-12-079060-2
- Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
- The permutations of periodic points in quadratic polynominials by J Leahy

## External links

*Algebraic solution of Mandelbrot orbital boundaries*by Donald D. Cross*Brown Method*by Robert P. Munafo- arXiv:hep-th/0501235v2 V.Dolotin, A.Morozov:
*Algebraic Geometry of Discrete Dynamics*. The case of one variable. - Gvozden Rukavina : Quadratic recurrence equations - exact explicit solution of period four fixed points functions in bifurcation diagram