Holomorphic Dynamics Part 1. Holomorphic dynamics on the Riemann sphere

Transcription

1 Holomorphic Dynamics Part 1. Holomorphic dynamics on the Riemann sphere In this part we consider holomorphic maps of the Riemann sphere onto itself. 1 Lyapunov stability. Fatou and Julia sets Here we give the definition of the Fatou and Julia sets in terms of the Lyapunov stability, and describe some elementary properties of these sets. 1.1 Global outlook There are natural questions that may be addressed to any class of dynamical systems. We will list some of them and discuss their relation to the holomorphic maps of the Riemann sphere. 1. Lyapunov stability properties. In our class, Lyapunov stability distinguishes so called Fatou and Julia sets. 2. Fractal properties of Julia sets. Holomorphic dynamical systems exhibit beautiful fractal sets that made the subject popular far beyond the professional mathematics community. 3. Number and structure of periodic orbits. 1

2 Poincaré said that periodic orbits, like torches, enlighten the properties of a dynamical system. The question is: what do the periodic orbits enlighten in holomorphic dynamics? 4. Complete description of the Fatou set. Progress of recent 30 years allowed us to describe the dynamics on the Fatou set in great details. 5. Hyperbolicity and structural stability. When the main features of a dynamical system persist under small perturbations, in other words, when the system is structurally stable? It appears that so called hyperbolicity is sufficient for this property to hold. I will try to organize the course in such a way that (almost) every lecture will have its own highlight, that is, a central idea that may go beyond the particular topics of the lecture. The prerequisite is the course on differential equations, but not necessary in dynamical systems, and a basic course in complex analysis. All the facts from these domains that are beyond the standard courses, will be explained. 1.2 Lyapunov stability The definition of the Lyapunov stability is well known, yet we present it here. Let X be a metric space with the distance d, and f : X X be a continuous map. Definition 1 A point p has a Lyapunov stable orbit under the map f, provided that for any ε > 0 there exists δ such that d(p, q) < δ implies d(f n (p), f n (q)) < ε for any n 0. An example is given by a rigid rotation of a plane: every point has a Lyapunov stable orbit. A natural question arises: is Lyapunov stability an open property? That is, is it true that the set of points whose orbits are Lyapunov stable, is open? The negative answer is justified by the following example. Consider a squeezed rotation of the plane: each circle centered at zero, is rotated by its own angle, For instance, z e i z z. The point zero has a Lyapunov stable orbit, but all other points do not. 2

3 1.3 Fatou and Julia sets In the majority of textbooks on holomorphic dynamics, the admission pass to the theory is not very cheep: for the mere understanding of the definition of the Fatou and Julia sets, one should pay by learning the properties of normal families of functions. Below we suggest an equivalent, but simpler definition. Definition 2 The Fatou set of a holomorphic map of the Riemann sphere onto itself is a set of all points that have Lyapunov stable orbits under the map. The Julia set is the complement to the Fatou set. 1.4 Opennes of Lyapunov stability in the holomorphic case In contrast to the above example, Lyapunov stability is an open property for holomorphic maps of compact manifolds. The main tool is the Cauchy estimate. We will prove the openness in dimension one. Theorem 1 The Fatou set is open. Proof Consider a holomorphic map f : Ĉ Ĉ, and a point p F (f). Let U be the δ neighborhood of p, where all the iterates f n, n 0, are defined and differ from f n (p) no more than by ε. Let { z R}, { z 1 } { }, R > 1, R be two charts on the Riemann sphere. Case 1. All the points f n (p) belong to the first chart { z R}, together with their ε-neighborhoods. Then f n (z) < R in U. In the δ/2-neighborhood V of p, all the iterates f n are Lipshitz with the constant L = 2R. Hence, for δ any q, q V, f n (q) f n (q ) < L q q. Hence, the orbit of any q V is Lyapunov stable. Case 2. Some of the points f(p) lie in the first chart { z R}, some in the second. We may repeat the previous arguments for all the points that belong to the first chart, next to those that belong to the second. This proves the Lyapunov stability of all points from some neighborhood of p, and thus openness of the Lyapunov stability property. 3

4 1.5 Rationality of holomorphic maps of the Riemann sphere Theorem 2 Any holomorphic map of the Riemann sphere to itself is rational. Proof Short version. Any map from the theorem is given by a meromorphic function. Any meromorphic function on Ĉ is rational. Detailed version. Consider two charts z C = Ĉ \ { } and w C = Ĉ \ {0}, z = 1. Let f : Ĉ Ĉ be a holomorphic map. This means that w in suitable charts f is given by holomorphic functions. Let p = f(q), p, q. Then z f is holomorphic as a function of z near q. Let p =, q. Then g = w f is holomorphic, g(q) = 0. The function g has zero of finite order at q. Hence, z f = 1 has a pole at q. w f If p, q =, then z f is holomorphic at infinity. If p = q =, then w f has zero, and z f has a pole at infinity. Therefore, z f is meromorphic on Ĉ, hence, rational. 1.6 Normal families and classical definition of the Fatou set Definition 3 A family F of holomorphic maps of a domain U Ĉ is called normal if for any sequence of maps of the family and any point p U there exists a neighborhood N of p and a subsequence of the sequence above that converges uniformly on N. The arguments in the proof of Theorem 1 immediately imply that the sequence of positive iterates of any rational map is normal on the Fatou set of this map. This implies the original definition of the Fatou set as the maximal set where all the positive iterates of the map form a normal family. The same argument imply that the family of holomorphic maps that are uniformly bounded on U is normal on U. Note, that, by the Ascoli-Arzela lemma, any sequence of a normal family on U admits a subssequence that converges uniformly on compact subsets of U. 4

5 1.7 Julia set is not vacuous A natural question arises: may be the Julia set vacuous? Same for Fatou sets The answer for the first question is negative. Theorem 3 The Julia set of a rational map of degree two or higher is nonvacuous. Proof Suppose, on the contrary, that for some rational f, J(f) =. Then the family of maps {f n n N} is normal on the whole Riemann sphere. But Ĉ is compact. Hence, by the Ascoi-Arzela lemma, there exists a subsequence f n k that converges uniformly on Ĉ. By the Weierstrass compactness theorem, the limit F of this subsequence is holomorphic. By Theorem 2, it is rational. Let d = deg f, and D = deg F. In this course we suppose, by default, that d > 1. We have that f n k uniformly tends to F. Then, for sufficiently large k, deg f n k = D. But deg f n k = (deg f) n k = d n k as k, a contradiction. 1.8 Fatou set may be vacuous Consider the next simple compact Riemann surface, after the sphere, namely the torus: T 2 = C/Z 2. A holomorphic map of C may be descended to the torus if it respects the factorization. Namely, any two points that differ by an integer vector are mapped to two points with the same property. A map G : C C, ζ 2ζ has this property. It may be descended to a map F : T 2 T 2. It is obvious that this map has no points with Lyapunov stable orbits: any two nearby points separate exponentially. In order to construct a map of a sphere with the same property it is sufficient to find a holomorphic function : T 2 C with the following property. The function will be two to one, and such that (ζ) = (ζ ) implies (2ζ) = (2ζ ). For instance, is even: ( ζ) = (ζ). Such a function exists; it is a famous Weierstrass function. The corresponding mapping f : Ĉ Ĉ satisfies the equation F = f. The map F is 4 to 1; hence, deg f = 4. This particular map f with the vacuous Fatou set was discovered by Schröder (1871) and rediscovered by Lattes (1918) in a more general context. This ends the naive approach to the study of Fatou and Julia sets. To get more serious results, we need elements of theory of Riemann surfaces, and of the Montel theory of normal families. 5