A BRIEF INTRODUCTION TO COMPLEX DYNAMICS DANNY STOLL Abstract. This paper discusses the theory of dynamical systems of a single variable on the complex plane. We begin with a brief overview of the general theory of holomorphic dynamics on complex manifolds, defining the Julia set and exploring its elementary properties. We then examine the local dynamics of rational maps near a fixed point, demonstrating the existence and uniqueness of the Böttcher map, a normal form for the dynamics of any differentiable map near a superattracting fixed point. We conclude with Douady and Hubbard’s proof of the connectedness of the Mandelbrot set. Contents 1. Introduction 1 2. Basic Theory 2 3. The Fatou and Julia sets 8 4. Local Fixed Point Theory: The Kœnigs and Böttcher Maps 13 5. The Quadratic Family 19 Acknowledgments 23 References 23 1. Introduction The study of iterated holomorphic functions on the complex plane dates back to Ernst Schröder’s work in 1870, but the field remains quite active and continues to expand, in part thanks to the development of computation. In 1870, Schröder first asked the question of whether or not an analytic function f fixing the point 0 and with derivative a = f 0(0) at the origin was conjugate to the linear map z 7! (f 0(0))z in a neighborhood of the origin. In 1884, Gabriel Kœnigs answered this question in the affirmative for 0 < jaj < 1. f n(z) Indeed (see Theorem 4.2), the function '(z) = limn!1 an explicitly solves the equation ' ◦ f = a'. Note that the case jaj > 1 follows trivially by iterating the local inverse of f. Thus began the study of iterated complex functions. The theory was further developed by the work of Auguste-Clémente Grévy, Lucyan Böttcher, and Leopold Leau, who studied the remaining cases jaj = 0 and jaj = 1. While the former case was answered completely by Böttcher, the latter was only answered by Leau in the parabolic case in which a is a root of unity, although the general case is now better understood due to the later work of Hubert Cremer and Carl Siegel, among others. Nevertheless, it is somewhat surprising that in all cases except the indefinite case jaj = 1, there is a single picture of local dynamics for all holomorphic functions. This is certainly not the case for real dynamical systems, and reflects just how strong a condition differentiability is for complex 1 2 DANNY STOLL functions (indeed, this has deep connections to the Maximum Modulus Principle; see, for instance, Shwartz’s Lemma 2.2). Pierre Fatou and Gaston Julia studied the global theory of iterated complex maps, defining the Fatou and Julia sets and proving a great number of their prop- erties despite the inability to see their shapes. Finally, thanks to the development of the study of dynamical systems in the later half of the 20th century as well as to advances in computer graphics, the stage was 2 set to study the quadratic family consisting of the maps fc(z) = z + c. Central to their study is the Mandelbrot set M of parameters c for which the Julia set of fc is connected. Adrien Douady and John Hubbard studied the quadratic family extensively, proving in 1980 that M is connected. Still, a number of questions remain open, most notably the conjecture that M is locally connected. This paper will lay out the foundations for the study of holomorphic dynamics. First, we explain a number of relevant results of complex analysis, such as the Uniformization Theorem (1.1) and the theory of covering spaces. We then begin our study of iterated functions, defining the Fatou and Julia sets and proving their basic properties, such as the density of iterated preimages in the Julia set J and the self-similarity of the geometry of J. Third, we examine the local dynamics of functions near a fixed point, construct- ing Kœnigs’ linearizing map near an attracting fixed point (Theorem 4.2) and the Böttcher map near a superattracting fixed point (Theorem 4.3). We conclude this section with the following theorem: Theorem 4.6. The Julia set J(f) for a polynomial f of degree d ≥ 2 is connected if and only if the filled Julia set K(f) contains every critical point of f. In this case, the complement C n K(f) is conformally isomorphic to the complement of the closed unit disk by the Böttcher map φ^, conjugating f to the map z 7! zd. If, instead, a critical point of f lies outside the filled Julia set, then K(f), and hence J(f), has uncountably many disconnected components. Remark 5.1. Hence the Julia set for the quadratic map f(z) = z2 +c is connected if and only if the critical orbit ff n(0) : n 2 Ng is a bounded subset of C. This theorem motivates the definition of the Mandelbrot set n M := fc 2 C : 9C; 8n ≥ 0; jfc (0)j < Cg: Finally, we present the proof of Douady and Hubbard’s famous theorem [DH82]: Theorem 5.3. The Mandelbrot set M is connected. 2. Basic Theory We begin with some basic definitions. If U ⊂ C is open, a function f : U ! C 0 0 f(z+h)−f(z) is called holomorphic if its derivative f : z 7! f (z) = limh!0 h is a continuous function from U to C. A standard result in complex analysis states that a holomorphic function is in fact analytic, in the sense that it is infinitely differentiable and equal to its power series in the neighborhood of a point. A holomorphic function is called conformal if its derivative is nowhere zero. (In some works, the term conformal also requires that a function be one-to-one; here, we use the term conformal isomorphism to indicate this additional property.) A BRIEF INTRODUCTION TO COMPLEX DYNAMICS 3 Henceforward, we will assume an understanding of the basic results of complex analysis (through [Con78], for example), though for convenience we will briefly summarize the major results used. A Riemann surface S is a complex one-dimensional manifold. That is, a topo- logical space S is a Riemann surface if for any point p 2 S, there is a neighborhood U of p and a local uniformizing parameter Φ: U ! C mapping U homeomorphically into an open subset of the complex plane. More- over, for any two such neighborhoods U and U 0 with nonempty intersection and local uniformizing parameters Φ and Ψ respectively, we require that Ψ ◦ Φ−1 be a holomorphic function on Φ(U \ U 0). Two Riemann surfaces S and S0 are said to be conformally isomorphic if there is a homeomorphism f : S ! S0 such that for neighborhoods U ⊂ S and U 0 = f(U) ⊂ S0 with local uniformizing parameters Φ and Ψ, the map Ψ ◦ f ◦ Φ−1 is conformal from Φ(U) to Ψ(U 0). One may check that this defines an equivalence relation on Riemann surfaces. A connected space X is simply connected if, roughly speaking, every loop in X can be contracted. More formally, X is simply connected if every continuous function f : S1 ! X from the unit circle to X is homotopic to a constant function, i.e. there exists a continuous function h : S1 × [0; 1] ! X with h(θ; 0) = f(θ) and h(θ; 1) = p for all θ 2 S1 and some p 2 X. A standard result in complex analysis follows. Theorem 2.1 (Uniformization Theorem). Let S be a simply connected Riemann surface. Then S is conformally isomorphic to either (1) the complex plane C, (2) the open disk D consisting of all z 2 C with absolute value jzj < 1, or (3) the Riemann sphere C^ consisting of C together with the point 1 with local 1 uniformizing parameter ζ(z) = z in a neighborhood of the point at infinity. The proof of Theorem ?? is rather deep, and is omitted in this paper. The following fact is also quite useful in our study. Lemma 2.2 (Schwarz’s Lemma). Let f : D ! D be a holomorphic map which fixes the origin. Then jf 0(0)j ≤ 1. If jf 0(0)j = 1, then f : z 7! zeiθ is a rotation about the origin. On the other hand, if jf 0(0)j < 1, then jf(z)j < jzj for all z 2 D n f0g. Proof. We apply the Maximum Modulus Principle, which asserts that any noncon- stant holomorphic function cannot attain its maximum absolute value over a region in the region’s interior. Notice that the quotient q(z) := f(z)=z is holomorphic on the disk and attains the value q(0) = f 0(0). Suppose first that q is nonconstant (if f(z) = cz, then clearly jcj ≤ 1). When jzj = r < 1, we have that jq(z)j < 1=r. Hence jq(z)j < 1=r whenever jzj ≤ r, since by the Maximum Modulus Principle, q cannot attain its maximum absolute value over the closed disk Dr of radius r on any point of the interior of this region. Letting r ! 1, we obtain jq(z)j ≤ 1 for all z in the disk. In particular, jq(0)j = jf 0(0)j ≤ 1. Applying the Maximum Modulus Principle again, we see that if jq(z)j = 1 for any z 2 D, then q(z) = c for all z, yielding jf 0(0)j = jq(0)j = 1 and contradicting the assumption that q is nonconstant. Thus if jf 0(0)j < 1 and q is nonconstant, then jf(z)j < jzj for all z 2 D n f0g, while if jf 0(0)j = 1, then q is constant.