Importance sampling

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 :

$m(f,z_0)=\lambda =\{\begin{array}{cc}{f_c}^{\prime }(z_0),& \text{if }{z}_0\ne \mathrm{\infty }\\ \frac{1}{{f_c}^{\prime }(z_0)},& \text{if }{z}_0=\mathrm{\infty }\end{array}$

where ${f_c}^{\prime }(z_0)$ 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 $A{x}^2+Bx+C=0$ 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 :

${P_c}^{\prime }(z)=\frac{d}{dz}{P}_c(z)=2z$

then

${P_c}^{\prime }({\alpha }_c)+{P_c}^{\prime }({\beta }_c)=2{\alpha }_c+2{\beta }_c=2({\alpha }_c+{\beta }_c)=2$

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 Mandelbrot 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 $ℂ$ to Riemann sphere (extended complex plane) $\widehat{ℂ}$ by adding infinity

$\widehat{ℂ}=ℂ\cup \{\mathrm{\infty }\}$

and extend polynomial $f_c$ such that $f_c(\mathrm{\infty })=\mathrm{\infty }$

Then infinity is :

• superattracting
• fixed point of polynomial $f_c$

$f_c(\mathrm{\infty })=\mathrm{\infty }=f_c^{-1}(\mathrm{\infty })$

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 $z^4+2c{z}^2-z+c^2+c=0$.

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 $x^4-A{x}^3+B{x}^2-Cx+D=0$ (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 $A^{\prime }=1,B=1,C=c+1$ 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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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

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• 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