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.