Complex plane maps

Maps of the complex plane can be much more complicated than real one-dimensional maps, but because they can be expressed as one-variable maps they retain some of the same properties, and are easier to characterize than fully two-dimensional real maps. Some of the most visually spectacular objects in dynamics arise in the study of complex maps, such as Julia sets and the Mandelbrot set. The mathematics of complex maps is rich enough that it is a research area unto itself, with many connections to other areas.

If $f(z)$ is a complex function of $z = x + i y$ , then we can think of it as a real 2-D function by taking the real and imaginary parts - i.e. if $f(z) = u(x,y) + i v(x,y)$ where $i^2=-1$ , then we get an associated 2D real map $g(x,y) = (u(x,y), v(x,y))$ . The two by two Jacobian matrix of $g$ then has a special structure:

$Dg = \left ( \begin{array}{cc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{array} \right ) = \left ( \begin{array}{cc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ -\frac{\partial u}{\partial y} & \frac{\partial u}{\partial x} \end{array} \right )$

i.e. the partial derivatives satisfy the Cauchy-Riemann equations $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ . This means that locally every complex map looks like a rotation and uniform scaling, which severely restricts the possible dynamics.

In this section we look at two of the historically important and foundational topics in complex maps: Newton's method for root-finding in the complex plane, and the extension of the quadratic map to the complex plane.

Newton's method for quadratics and cubics

In the Introduction we mentioned that Arthur Cayley's 1879 investigation of Newton's method for complex roots of polynomials was one of the first intricate dynamical systems to be studied.

Recall that Newton's method aims to iteratively solve the equation $f(x) = 0$ starting at $x_0$ by approximating $f(x)$ with its tangent line (with the tangency point being $x_0$ ), which gives the formula

$x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}$

Quadratics

Lets start (as Cayley did) with the much simpler case of roots of quadratic polynomials. Of course in this case we have the quadratic formula and it is easy to directly compute the roots. For

$p(z) = z^2 + b z + c = (z - r_1)(z - r_2) = 0$

Newton's method gives us the iterative formula

$z_{k+1} = N(z_k) = z_k - \frac{p(z_k)}{p'(z_k)} = z_k - \frac{z_k^2 + b z_k + c}{2 z_k + b} = \frac{z_k^2 - c}{2 z_k + b}$

The last form shows that if $z_k = -b/2 = \frac{r_1 + r_2}{2}$ , at the midpoint of the roots, the next iterate is undefined.

To see the rest of the dynamics of this map clearly it helps a lot to conjugate it to a simpler form. One choice is to see if all quadratic Newton maps are conjugate to the case where $r_1 = 1$ and $r_2 = -1$ , in which $p = z^2 - 1$ and

$z_{k+1} = \frac{z_k^2 + 1}{2 z_k}$

This is possible by using the linear conjugacy

$h = 1 - \frac{2 (z-r_1)}{r_2-r_1} = \frac{2 z - r_1 - r_2}{r_1 - r_2} = \frac{2 z + b}{\sqrt{b^2-4c}}$

as long as $r_1 \neq r_2$ . The case of a double root must be analyzed separately.

The map $N_s = \frac{z^2 + 1}{2 z}$ can be further simplified with a second conjugacy that sends $1$ to $0$ , and $-1$ to infinity:

$g(z) = \frac{z-1}{z+1}$

Then $g(N_s(g^{-1}(z))) = z^2$ , so Newton iteration for any quadratic with distinct roots is conjugate to the map $z^2$ .

To analyze $z^2$ in the complex plane it helps to use the polar exponential form $z = r e^{i \theta}$ where $i^2 = -1$ and $r \ge 0$ . Then $z^2 = r^2 e^{2 i \theta}$ , which means the radius $r$ is squared and the angle $\theta$ doubled at every iteration. The iterates of any point inside the unit circle (with $r<1$ ) will approach $0$ , and the points outside the unit circle will diverge to infinity. Points on the unit circle will stay on the unit circle, with the dynamics of the doubling map $\theta_{i+1} = 2 \theta_i (\text{mod } 2 \pi)$ .

This means the dynamics of the map $N_s$ are quite simple, with points in the right half-plane converging to $1$ and points in the left half-plane converging to $-1$ . Points on the imaginary axis will be transformed by a map conjugate to the doubling map.

Cubics

For the general cubic equation $p(z) = z^3 + a z^2 + b z + c$ , Newton's method is

$z_{k+1} = z_k - \frac{p(z_k)}{p'(z_k)} = \frac{2 z_k^3 + a z_k^2 - c}{3 z_k^2 + 2 a z_k + b}$

Cayley looked at one of the simplest cases, three roots of unity $z^3 - 1 = 0$ , for which Newton's method simplifies to

$z_{i+1} = \frac{2}{3}z_i + \frac{1}{3 z_i^2}$

He was unable to completely analyze the basins of attraction for that system, which in hindsight is not surprising since the three basins have a complicated fractal boundary (see the applet below).

For some cubics, there are open sets in which Newton's method fails to converge.

Julia and Fatou sets

Quadratic maps can be put into the form

$z_{i+1} = z_i^2 + c$

using a linear conjugacy (instead of, for example, the logistic map form $z_{i+1} = \lambda z_i (1 - z_i)$ ). This has long been used as the standard form in the complex variable case. (Its a good exercise to calculate what that conjugacy is, i.e. $h(z) = Az+B$ and we want $h(z^2+c) = \lambda h(z) (1 - h(z))$ .)

For a fixed value of the parameter $c$ , we can study the behavior of the iterated system for different initial values of $z$ . Already back in the early 1900s, Pierre Fatou, Gaston Julia, and Samuel Lattes were able to prove some interesting theorems about this behavior. Using some techniques of complex analysis (primarily Montel's theory of normal families of functions), they were able to show that there are three types of dynamics for different initial values: either the orbit diverges to infinity, or it converges to a unique attracting fixed point, or it stays within an invariant set called the Julia set. The Julia set is the closure of the set of repelling periodic points of the map (several other equivalent definitions have been used).

The Mandelbrot set

The quadratic map $z_{i+1} = z_i^2 + c$ is a family with two complex parameters, $c$ and $z_0$ . For Julia sets we consider $c$ to be fixed, and examine the dynamics as a function of $z_0$ . It turns out that the orbit of the critical points of a map is espectially important for these and other complex maps, so we can alternatively study the dynamics of the critical points as a function of $c$ . The quadratic map only has one critical point, $z=0$ , so we can study it in a single plane of $c$ values. The set of $c$ values for which the iterates of $z_0=0$ do not diverge is called the Mandelbrot set.

Montel's theorem

In real analysis, an important result is the Arzela-Ascoli theorem, which provides necessary and sufficient conditions for an infinite family of functions to have uniformly convergent subsequences. One of the conditions is that the family is uniformly equicontinuous, meaning that for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $w$ and $z$ with $|w - z| < \delta$ , $|f(w) - f(z)| < \epsilon$ for every member of the family. The fact that $\delta$ is independent of the locations $w$ and $z$ is the "uniform" part, and that it is also independent of which family member is the "equi" part.

For families of complex analytic functions, such as polynomials and rational functions, fewer conditions are needed to guarantee the existence of uniformly convergent subsequences. Various theorems about this are (confusingly) called Montel's theorem. One of the versions of Montel's theorem(s) is: if $\mathcal{F}$ is a family of analytic functions defined on an open set $\Omega \subset \mathbb{C}$ , and it is uniformly bounded on every compact subset of $\Omega$ , then $\mathcal{F}$ is normal, where in this context normal means that every sequence of functions in $\mathcal{F}$ has a convergent subsequence that converges uniformly on any compact subset of $\Omega$ .

One of the strongest versions of Montel's theorem is that if a family of complex functions omits two values, i.e. if there are complex values $a$ and $b$ such that $f_m(z) \neq a$ and $f_m(z) \neq b$ for all $f_m \in \mathcal{F}$ , then it is a normal family.

It is beyond the scope of this work to include the necessary complex analysis to prove this: the Julia set of a rational map is the set of points where the family of iterates of the map is not normal. This means that on any neighborhood of a point of the Julia set, the iterates of that neighborhood will eventually cover all of the complex plane (apart from maybe 1 value). This result is why the Julia set boundary of the super-attracting root basins of Newton's method always contain portions of every root's basin of attraction.

Moebius transformations

Fractional linear transformations, or Moebius transformations, came up earlier in this text in the context of the Schwartzian derivative (they are the only functions with vanishing Schwartzian). In complex analysis and dynamics they play an important role, so we will consider a few more of their most relevant properties.

We will exclude constant maps from what we will call Moebius transformations - let us define a Moebius transformation as a complex function of the form

$$ A(z) = \frac{A_{1,1} z + A_{1,2}}{A_{2,1} z + A_{2,2}} $$ with complex parameters $A_{i,j}$ such that

$A_{1,1} A_{2,2} - A_{2,1} A_{1,2} \neq 0$

If we compose two Moebius transformations we find the result is also a Moebius transformation, and more surprisingly the composition operation is identical to matrix multiplication if we consider the parameters as matrix entries. This means that Moebius transformations form a group under composition which is almost identical to the multiplicative group of two by two matrices with nonzero determinant. There is an important difference, which is that the parameters of a Moebius transformations can be rescaled by a nonzero complex constant without changing the function. So the group of Moebius transformations is isomorphic to what is called the projective general linear group $PSL(2,\mathbb{C})$ .

One handy corollary of the above information is that we can write down a simple formula for the inverse of a Moebius transformation:

$A^{-1} = \frac{A_{2,2} z - A_{2,1}}{-A_{1,2} z + A_{1,1}}$

By considering the fixed-point equation $A(z) = z$ we see that there are two solutions; if they are not distinct the function is called parabolic, and it is conjugate to a translation $B z + D$ (the conjugacy also being a Moebius transformation).

Exercises

Calculate the fixed points and their stability for the complex map $f(z) = z^2 + z - 4$ .
Calculate the fixed points and their stability for the conjugate map

$g(z) = h^{-1} \circ f \circ h(z)$

where $h(z)$ is the Riemann sphere inversion $1/z$ .
For the complex map $f(z) = z^3$ :
- Find all initial points whose trajectories are bounded and do not converge to the origin.
- Find the Lyapunov exponents of all bounded orbits (hint: it helps to work in polar coordinates).
- Show that $f$ has chaotic orbits
Find and classify the stability of the period-2 orbit of the map $f(z) = 1 + iz^2$ .
For the quadratic map $Q_c(z) = z^2 + c$ , show that the period-2 bulb of the Mandelbrot set is a disk. (The period-2 bulb is defined to be the set of complex $c$ such that there is an attracting period-2 orbit for $Q_c$ .)