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

$f_{c}(z)=z^{2}+c\,$

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.

$\ f_{c}^{(k)}(z)$ is the $\ k$ -fold compositions of $f_{c}\,$ with itself = iteration of function $f_{c}\,$ or,

$\ f_{c}^{(k)}(z)=f_{c}(f_{c}^{(k-1)}(z))$

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

$\ z:f_{c}^{(p)}(z)=z$

where $\ p$ is the smallest positive integer.

We can introduce a new function:

$\ F_{p}(z,f)=f_{c}^{(p)}(z)-z$

so periodic points are zeros of function $\ F_{p}(z,f)$ :

$\ z:F_{p}(z,f)=0$

which is a polynomial of degree $\ =2^{p}$

Stability of periodic points (orbit) - multiplier

Stability index of periodic points along horizontal axis

boundaries of regions of parameter plane with attracting orbit of periods 1-6

Critical orbit of discrete dynamical system based on complex quadratic polynomial. It tends to weakly attracting fixed point with abs(multiplier)=0.99993612384259

The multiplier ( or eigenvalue, derivative ) $m(f,z_{0})=\lambda \,$ of rational map $f\,$ at fixed point $z_{0}\,$ is defined as :

$m(f,z_{0})=\lambda ={\begin{cases}f_{c}'(z_{0}),&{\mbox{if }}z_{0}\neq \infty \\{\frac {1}{f_{c}'(z_{0})}},&{\mbox{if }}z_{0}=\infty \end{cases}}$

where $f_{c}'(z_{0})\,$ is first derivative of $\ f_{c}$ with respect to $z\,$ at $z_{0}\,$.

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

Multiplier is:

• 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 : $abs(\lambda )\,$

Periodic point is :

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

Where do periodic points belong?

• attracting is always in Fatou set
• repelling is in the Julia set
• Indifferent fixed points may be in the one or in the other. 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 $f$. 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 (both repelling)

Complex dynamics

Fixed points for c along horizontal axis

Fatou set for F(z)=z*z with marked fixed point

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 $\lambda _{\beta _{c}}=1+{\sqrt {1-4c}}\,$

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).
• $\alpha _{c}\,$ 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

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

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 of polynomial $f_{c}\,$

$f_{c}(\infty )=\infty =f_{c}^{-1}(\infty )\,$

Period-2 cycles

Bifurcation from period 1 to 2 for complex quadratic map

Suppose next that we wish to look at period-2 cycles. That is, we want to find two points $\beta _{1}$ and $\beta _{2}$ such that $f_{c}(\beta _{1})=\beta _{2}$, and $f_{c}(\beta _{2})=\beta _{1}$.

Let us start by writing $f_{c}(f_{c}(\beta _{n}))=\beta _{n}$, and see where trying to solve this leads.

$f_{c}(f_{c}(z))=(z^{2}+c)^{2}+c=z^{4}+2z^{2}c+c^{2}+c.\,$

Thus, the equation we wish to solve is actually $z^{4}+2cz^{2}-z+c^{2}+c=0$.

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 $\alpha _{1}$ and $\alpha _{2}$, 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 :

first method

$(z-\alpha _{1})(z-\alpha _{2})(z-\beta _{1})(z-\beta _{2})=0.\,$

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 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 $D=c\beta _{1}\beta _{2}$ 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 (the formulas are a tad messy) that :

$f_{c}(\beta _{1})=\beta _{2}$ and $f_{c}(\beta _{2})=\beta _{1}$

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

Second method of factorization

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

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.

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 $|\beta _{1}|=|\beta _{2}|=1$. 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>2

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

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