Chapter 8 Complex Dynamics

In the last part of this course we study the dynamics of complex maps f : C → C. Here we find the richness of dynamics in R2, but in a context where we can prove much more, thanks to the powerful and beautiful machinery of complex analysis.

The history of the subject is remarkable. In the decade following the end of the First World War in 1918 the French mathematicians Pierre Fatou and Gaston Julia developed the theory of iterated rational maps (maps of the form z → P (z)/Q(z) where P and Q are polynomials). They did so independently and very much in competition, proving a very impressive body of results but leaving unsolved some central questions. The subject went into a quiet period - as often happens in mathematics after a wave of new development - for about 50 years, until around 1980 a number of leading present day mathematicians (Mandelbrot, Douady, Hubbard, Sullivan, Milnor, Thurston) began taking up the questions where Fatou and Julia had left off, with inspiration provided by the extraordinarily beautiful pictures produced by computer graphics. The subject has given birth to to remarkable pictures and to remarkable mathematics. Moreover, as we shall see, this mathematics tells us much more about the real one-dimensional maps we looked at earlier in the courses - in particular the logistic family, period-doubling etc.

We first recall some basic facts and theorems from complex analysis.

Definitions

A map f : C → C is said to be differentiable at z0 if f(z) − f(z ) lim 0 z→z0 z − z0

0 exists, in which case we call the value of this limit the derivative f (z0) of f at z0.

f is said to be holomorphic at z0 if f is differentiable at every point z in some neighbourhood of z0 (i.e. there exists a real number r > 0 such that f is differentiable at every z such that |z − z0| < r).

f is said to be analytic at z0 if there exist a real number r > 0 and complex numbers an (1 ≤ n < ∞) such that X n f(z) = an(z − z0) ∀z with |z − z0| < r n=0

1 Key Property (which can be proved using Cauchy’s theorem):

A map f : C → C is analytic at z0 ⇔ f is holomorphic at z0.

Being analytic is a very strong property indeed: it says that once we know the Taylor series of f at z0 we know the exact value of f(z) at every point z inside the disc of convergence of the Taylor series - the remainder term that we have in the real form of Taylor’s Theorem is zero in the complex case.

0 Geometrical interpretation of f (z0)

0 If f is differentiable at z0, with derivative f (z0), then, to a first approximation

0 f(z) − f(z0) = (z − z0)f (z0)

So, to a first approximation, f acts on a small round disc centred on z0 by expanding it by a scale factor 0 0 |f (z0)|, turning it through an angle arg(f (z0)), and centring the resulting disc on f(z0).

0 Note that it follows that f is conformal (angle-preserving) at z0 if f (z0) 6= 0.

If f is differentiable at z0 and z0 is a fixed point of f, then it is self-evident from this geometrical picture that: 0 z0 is an attractor if |f (z0)| < 1

0 z0 is a repeller if |f (z0)| > 1

0 z0 is neutral if |f (z0)| = 1

Example: f(z) = z2 − 1 √ √ √ This has fixed points z = (1 ± 5)/2. The derivative of f is 2z. Since both |1 + 5| > 1 and |1 − 5| > 1 both fixed points are repellers.

If we are concerned with a periodic orbit z0 → z1 → z2 → ... → zn = z0 then, to a first approximation, 0 0 the first map is (locally) multiplication by f (z0), and the second is multiplication by f (z1) etc. Hence the n 0 0 0 composite f is (locally) multiplication by the product f (z0)f (z1) . . . f (zn−1) = ζ, called the multiplier n n of the orbit. (Note that by the chain rule ζ is the derivative of f at z0, or indeed the derivative of f

at any other point of the cycle {z0, . . . , zn−1}.)

This period cycle is: attracting if |ζ| < 1

repelling if |ζ| > 1

neutral if |ζ| = 1

2 Example: f(z) = z2 − 1 The period two cycle 0 → −1 → 0 is an attractor since it has multiplier

0 × (−2) = 0. An attractor with multiplier 0 is called a superattractor.

We shall concentrate on polynomial maps, and in particular on quadratic maps

2 f(z) = αz + βz + γ (α, β, γ ∈ C, α 6= 0)

This looks like a 3-parameter family of maps, but we can reduce the number of parameters if we recall that conjugate maps have the same dynamics.

Definition

We say that analytic maps f and g from C to C are conjugate if there exists an analytic bijection h : C → C, with analytic inverse, such that g = hfh−1.

Fact (from complex analysis): h : C → C is an analytic map, with an analytic inverse, if and only if h has the form h(z) = kz + l for some k, l ∈ C with k 6= 0.

Proposition

Every quadratic map f(z) = αz2 + βz + γ (α, β, γ ∈ C, α 6= 0) is conjugate to a unique map of the form 2 qc(z) = z + c (c ∈ C).

Proof

We try to find a map h(z) = kz + l such that hf(z) = qch(z)∀z ∈ C.

hf(z) = k(αz2 + βz + γ) + l

2 qch(z) = (kz + l) + c

2 2 Equating coefficients of z ,z and 1 in these expressions we see that hf = qch if and only if kα = k , kβ = 2kl and kγ + l = l2 + c.

Thus for a given α 6= 0, β and γ we may take k = α, l = β/2 and get a conjugacy h(z) = kz + l from f to 2 qc where c = αγ + β/2 − β /4. Indeed since we ask for h to be an analytic bijection with analytic inverse it must have the form h(z) = kz + l, so for a given α 6= 0, β and γ we obtain a unique value of c ∈ C. 

Remark The same proof goes through in the real case to show that every x → αx2 + βx + γ with

α 6= 0, β, γ ∈ R is conjugate to some x → x2 + c with c ∈ R, but we can no longer deduce immediately that the value of c is unique, since there are many diffeomorphisms of R which are not of the form x → kx + l.

3 We have just shown that every quadratic polynomial is conjugate to one of the form z → z2 + c. Equally well we can prove that every quadratic polynomial is conjugate to one in the logistic family

fµ(z) = µz(1 − z) (where now µ ∈ C):

Proposition 2 qc is conjugate to fµ ⇔ c = µ/2 − µ /4.

Proof

This can be proved from first principles (like the previous proposition), or more simply by applying the

previous proposition with α = −µ, β = µ and γ = 0. 

Note that for most values of c there are two values of µ with fµ conjugate to qc, the exception being c = 1/4, for which there is just one value of µ, namely µ = 1.

Corollary (of the proposition immediately above)

2 qc has an attracting fixed point if and only if c = µ/2 − µ /4 for some µ with |µ| < 1.

The fixed points of fµ are 0 and 1−1/µ. These have multipliers µ and µ−2µ(1−1/µ) = 2−µ respectively. 2 So qc has an attracting fixed point if and only if c = µ−µ /4 for some µ with either |µ| < 1 or |2−µ| < 1. 2 2 2 But µ/2 − µ /4 = (2 − µ)/2 − (2 − µ) /4. So qc has an attracting fixed point if and only if c = µ/2 − µ /4 for some µ with |µ| < 1. 

The set {c ∈ C : c = µ/2 − µ2/4 for some µ with |µ| < 1} is a cardioid, which intersects the real axis in the open interval (−3/4, 1/4).

2 Dynamics of qc : z → z + c

2 When c = 0 this is the map q0 : z → z .

In this case initial points z0 with |z0| < 1 have orbits tending to the super-attracting fixed point 0.

Initial points z0 with |z0| > 1 have orbits tending to ∞ (which can also be thought of as a super-attracting fixed point, as we shall see).

On the unit circle the map q0 behaves as the doubling map (the binary shift) t → 2t mod 1 (where arg(z) = 2πt, i.e. z = e2πit), discussed earlier in the course.

Thus on the unit circle q0 has sensitive dependence on initial conditions, and on this circle there is a sense collection of repelling periodic orbits (constructed earlier in the course): m 2nt = t mod 1 ⇔ 2nt − t = m some m ∈ ⇔ t = Z 2n − 1

4 The point 0 is a critical point for q0 (that it, a point where the derivative of q0 is zero). Earlier we saw the geometrical interpretation of a non-zero derivative. We now consider the geometrical interpretation 0 of the fact that q0(0) = 0. A small disc of radius r around z = 0 is mapped by q0 to a much smaller disc, of radius r2. This map is two-to-one, and the boundary of the first disc is mapped twice around the smaller disc. The map is not conformal at the critical point 0. It doubles angles there.

2 Returning to our study of the family qc : z → z + c, if we perturb c just a little way away from 0 we find the following dynamics. There is still an attracting fixed point, but it is now a little way away from the critical point 0 (the fixed point is no longer super-attracting). The complex plane is divided into:

(i) the basin of attraction of the attracting fixed point z0; (ii) the basin of attraction of ∞;

(iii) the complement of (i) and (ii), a fractal curve homeomorphic to a circle, called the Julia set of qc.

For any small value of c, there is a homeomorphism from the Julia set of qc to the unit circle, conjugating the action of qc on this Julia set to the action of q0 on the unit circle. But for larger values of c the Julia set becomes more and more distorted and eventually collides with itself and breaks up into a Cantor set. But before we look at this process in detail we need a formal definition of the Julia set of a polynomial.

For this it is convenient to regard ∞ as a point like any other point in C.

The Riemann sphere

Consider the complex plane as the plane C = {x + iy : x, y ∈ R} = {(x, y, 0) : x, y ∈ R} ⊂ R3. Let S2 denote the unit sphere in R3. Thus the intersection of S2 with the complex plane is the equator circle of S2.

Now define a bijection π : S2 → C ∪ {∞} by stereographic projection from the ‘north pole’ of S2, the point N = (0, 0, 1) ∈ R3 (send the point N itself to ∞). Define the distance d(z, w) between any two points of C ∪ {∞} to be the distance on S2 along a great circle arc between π−1(z) and π−1(w). This is called the distance in the spherical metric between z and w. Cˆ = C ∪ {∞} is known as the Riemann sphere. From now on we shall use the spherical metric on the Riemann sphere.

Facts

(i) The complex differentiable maps Cˆ → Cˆ are the rational maps P (z) z → Q(z)

(where P and Q are polynomials).

5 (ii) The invertible maps among these are the M¨obius transformations

az + b z → cz + d

(where ad − bc 6= 0).

Let f be a complex differentiable map Cˆ → Cˆ.

Definitions

The Fatou set F (f) is defined by:

n z0 ∈ F (f) ⇔ {f }n>0 are equicontinuous at z0

n n (i.e. given  > 0, ∃δ > 0 such that, ∀n > 0, d(f (z), f (z0)) <  whenever d(z, z0) < δ, where d(z, z0) denotes distance in the spherical metric).

The Julia set J(f) is the complement of F (f) in Cˆ.

Thus the Julia set is the set of points z0 such that f has sensitive dependence on initial conditions at z0.

Example

For f(z) = z2 we have F (f) = {z : |z|= 6 1}

J(f) = {z : |z| = 1}

Useful properties of F (f) and J(f) (without proof)

1. F (f) is open (this is immediate from the definition) and so J(f) is closed (that is to say J(f) contains all its limit points).

2. F (f) is completely invariant i.e. f(F (f)) ⊆ F (f) and f −1(F (f)) ⊆ F (f); hence J(f) is also completely invariant.

3. J(f) is the closure of the set of all repelling periodic points of f (the was the original definition of J(f)).

There are many more results that can be proved about F (f) and J(f) (for example: any closed completely

invariant set containing at least three points must be either J(f) or Cˆ). See Beardon’s book ‘Iteration of rational functions’ for statements and proofs of the key theorems of Fatou-Julia theory. But we now specialise back to polynomials, where ∞ has a special role.

6 Definition

The filled Julia set K(P ) of a polynomial P is the set:

n K(P ) = {z0 : P (z0) does not tend to ∞ as n → ∞}

Another useful property

4. The Julia set of P is the boundary of K(P ), i.e.

J(P ) = {z0 ∈ K(P ): ∀ > 0 ∃ z with |z − z0| <  and z∈ / K(P )}

Example

For P (z) = z2 we have K(P ) = {z : |z| ≤ 1}

J(P ) = {z : |z| = 1}

Note that the unit circle is indeed the closure of the set of repelling periodic points (the points of the

n form e2πmi/(2 −1)), in agreement with our useful property 3.

Practical ways of plotting Julia sets

2 −1 If f is a rational map of degree > 1 (for example qc : z → z + c) then f is a multi-valued map (e.g. −1 √ qc : z → (z −c) is 2-valued). As the Julia set contains a dense set of repelling periodic orbits, iterating f from a starting point near to J(f) sends you away from J(f). But iterating the branches of f −1 sends you towards J(f). We regard the branches of f −1 as an iterated function system and use them to plot J(f) in just the same way that used such systems in the previous chapter to plot fractals.

Algorithm

−1 −1 Take as initial point almost any z0 ∈ C. Plot all the values of f (z0), then all the values of f applied to these points etc. The set you plot will accumulate on J(f).

−1 −1 Alternatively just pick one of the values of f (z0) at random and plot it, then one of the values of f applied to this new point etc.

If the initial point z0 is not on J(f) then to get a picture of J(f) with no extraneous points you should not plot the first few hundred points you calculate by this process.

However it is usually possible to start the process at a point z0 which is known to be in J(f) (for example any repelling fixed point is necessarily in J(f)) and in this case every point calculated will be on J(f), since J(f) is completely invariant.

7 Remarks

1. The only points that must not be used as initial points in this process are points with finite backwards orbits (for example the point z = 0 for the map f(z) = z2).

2. There are refinements to the algorithm to make sure points in different parts of J(f) are equally likely to be plotted, but the crude algorithm often works quite well.

Topology of J(qc) as c varies

2 Recall that qc denotes the polynomial z → z + c. For c in the cardioid where qc has an attracting fixed point we have seen that the dynamics of qc resembles that of qc. In particular the Julia set J(qc) is homeomorphic to the unit circle and restricted to J(qc) the map qc is conjugate to the doubling map on the circle.

But as c approaches the boundary of the cardioid the Julia set of qc becomes an increasingly distorted circle. We next consider the dynamics of qc at c = −1 (which lies outside the cardioid).

2 Dynamics of q−1 : z → z − 1

2 √ q−1 : z → z − 1 has two fixed points, z = (1 ± 5)/2 (= 1.618 and −0.618), both of which are repelling √ √ since |1 + 5| > 1 and |1 − 5| > 1. It has a period 2 cycle {0, −1} which is super-attracting, since this orbit has multiplier 0 × (−2) = 0.

The Fatou set consists of a countable infinity of components which are each homeomorphic to a round disc (see pictures handed out during the course). We examine how these components map to one another under q−1. The first two such components F1 and F2 surround the two points 0 and −1 respectively of the 2-cycle, and the boundaries of these components touch at the fixed point −0.618. The map q−1 exchanges F1 with F2, sending F1 two-to-one onto F2 (since z = 0 is a critical point, near which the map 2 is the two-to-one map z → z ) and sending F2 one-to-one onto F1. There is another component which maps one-to-one onto F1. This component, which we shall denote F3, lies on the other side of F1 from

F2. It is centred on z = +1 and consists of all points z such that −z lies in F2. Next there are two components which each map one-to-one onto F3, then for each of these there two components mapping onto them and so on. It can be proved that all the components of the Fatou set are preperiodic and are eventually mapped onto the two cycle of components {F1,F2} (except for the one component containing ∞, which of course is mapped to itself.) Fatou conjectured in the 1920s that every component of the Fatou set of a rational map is preperiodic. This was proved by the American mathematician Dennis Sullivan in 1985 as the celebrated ‘No wandering domains’ Theorem, which was one of the results which restarted interest in the subject of iterated rational maps 60 years on from the work of Fatou and Julia.

8 The transition between q0 and q−1

2πi/3 3πi/3 The map q0 has an attracting fixed point at 0 and a repelling 2-cycle {e , e }. As we move c along the real axis from c = 0 towards c = −1 these three points move closer together until at the value c = −3/4 they collide, to form a single neutral fixed point (having multiplier −1), after which they ‘exchange identities’ and separate as a repelling fixed point and an attracting 2-cycle. This collision is what alters the topology of J(qc), as all the inverse images of the colliding points also collide in pairs.

Note that c = −3/4 corresponds to µ = 3 for the logistic family fµ(x) = µx(1 − x): if we look at qc restricted to the real axis, what we see as c moves through the value c = −3/4 is a pitchfork bifurcation of the attracting fixed point into a repelling fixed point and an attracting 2-cycle. This is exactly the same bifurcation that we saw for the logistic map at µ = 3, but now viewed in the complex plane, not just in the real ‘slice’.

Values of c for which qc has an attracting 2-cycle

Period 2 cycles and fixed points of qc are solutions of qc(qc(z)) = z, that is:

2 4 2 2 2 2 (z2 + c) + c − z = 0 ⇔ z + 2cz − z + c + c = 0 ⇔ (z − z + c)(z + z + c + 1) = 0

The fixed points are the solutions of z2 − z + c = 0 so there is at most one prime period 2 orbit, and this consists of the solutions {u, v} of z2 + z + c + 1 = 0, where

u + v = −1 and uv = c + 1

The multiplier of this orbit is q0(u)q0(v) = 4uv = 4(c + 1)

Thus the 2-cycle is an attractor if and only if

|c + 1| < 1/4 that is to say, if an only if c lies in an open round disc, having centre −1 and radius 1/4. Observe that the boundaries of the cardioid and this disc touch at z = −3/4.

The Mandelbrot set

Continuing along the real axis in the plane of the complex parameter c, we find a sequence of smaller and small topological discs (no longer round) with the property that qc has an attracting 4-cycle, an attracting 8-cycle, etc (corresponding to the period-doubling cascade for real maps, that we investigated

9 earlier). This sequence of discs ends at the Feigenbaum point. Beyond that there are a whole string of further small discs corresponding to attracting orbits of periods which are not powers of 2, the most prominent of these being the one corresponding to period 3. Eventually, at c = 2 we reach the end point of the Mandelbrot set (defined below). The Julia set of q−2 is the closed interval of real axis [−2, +2] (see Exercise Sheet 9).

There are other interesting regions of parameter space off the the real axis. For example there are two topological discs attached to the cardioid which have the property that if c lies in either of them then the map qc has an attracting 3-cycle. But it can be shown that for any c with |c| large enough, the Julia set of qc is a Cantor set.

Definition

The Mandelbrot Set M is the set of all values of c ∈ C such that the Julia set J(qc) is connected. Equivalently n M = {c ∈ C :(qc) (0) does not tend to ∞ as n → ∞}

This equivalent definition is used in the computer algorithms which are used to draw pictures of M.

For c∈ / M the Julia set of qc is a Cantor set. (For a proof of this see, for example, Devaney’s book.)

Historical Note. The first published picture of what is now called the Mandelbrot Set appeared in a paper by Brooks and Matelski in 1978, in the context of investigating discreteness criteria for Kleinian groups, but it was a very primitive picture, plotted with ‘x’s on a line printer. Mandelbrot independently started studying z → z2 + c in about 1980.

The structure of M is very intricate - arbitrarily close to any point of the boundary of M there is a small copy of the whole of M (and one can keep zooming and zooming forever) - for example if one zooms on the small period 3 component on the real axis one sees a whole copy of M in which this period 3 component corresponds to the cardioid. Mandelbrot’s first pictures suggested that M had infinitely many ‘offshore islands’, but then Douady and Hubbard showed that all these ‘islands’ must be joined to the mainland by ‘filaments’, by proving the following theorem:

Theorem (Douady and Hubbard 1982): M is connected.

In 1982 Douady and Hubbard also made two conjectures, which are still uproved today:

Conjecture 1

c ∈ interior M ⇔ qc has an attracting periodic cycle

10 This is the conjecture that every component of the interior of M is like the ones we have investigated (the cardioid, and the disc with centre −1 and radius 1/4), and that there are no ‘strange’ components.

Conjecture 2 M is locally connected

This says that given any c ∈ M and open set U in C containing c, there exists an open set V ⊂ U, still containing c, such that M ∩ V is connected.

Douady and Hubbard proved that Conjecture 2 implies Conjecture 1. If Conjecture 2 could be proved we could deduce a complete combinatorial description of M in terms of binary sequences, and many leading mathematicians have worked very hard on this conjecture for the last 25 years, proving it for a larger and larger class of points c: in this period two mathematicians (Yoccoz and McMullen) who have worked on this problem have been awarded Fields Medals. There might seem every reason to believe it to be true, except for the facts that (a) the analogue of the Mandelbrot set for cubics is known to be non-locally connected, and (b) there are many examples of quadratic Julia sets which are not locally connected.

There are many interesting Julia sets to be be found by choosing appropriate values of c ∈ M. For example, if z = 0 is strictly preperiodic for qc then J(qc) is a dendrite (a ‘tree-like’ set). Examples of values of c where z = 0 is preperiodic are c = i and c = −2 (for c = −2 the Julia set is the real interval [−2. + 2]): z → z2 + i : 0 → i → −1 + i → −i → −1 + i → −i → ...

z → z2 − 2 : 0 → −2 → +2 → +2 → ...

For more on Julia Sets and the Mandelbrot Set see Alan Beardon’s book Iteration of Rational Functions or John Milnor’s book Dynamics in One Complex Variable. The deeper you go into this subject the more fascinating it gets, both from the point of view of the mathematics and of the pictures. See Devaney’s web-site for programs for drawing Julia Sets and the Mandelbrot Set or just look around the web using your favourite search-engine.

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