Dynamics of Entire Functions

Complex dynamics of iterated entire holomorphic functions is an active and exciting area of research. This manuscript collects known background in this field and describes several of the most active research areas within the dynamics of entire functions. Complex dynamics, in the sense of holomorphic iteration theory, has been a most active research area for the last three decades. A number of interesting developments have taken place during this time. After the foundational work by Fatou and Julia in the early 20th century, which developed much of the basic theory of iterated general rational (and also transcendental maps), the advent of computer graphics D. Schleicher School of Engineering and Science, Research I, Jacobs University, Postfach 750 561, D-28725 Bremen, Germany e-mail: [email protected] G. Gentili et al. (eds.), Holomorphic Dynamical Systems, 295 Lecture Notes in Mathematics 1998, DOI 10.1007/978-3-642-13171-4 5, c © Springer-Verlag Berlin Heidelberg 2010 296 Dierk Schleicher made possible detailed studies of particular maps; most often, quadratic polynomials as the simplest non-trivial holomorphic mappings were studied. While a number of deep questions on quadratic polynomials remain, interest expanded to specific (usually complex oneor twodimensional) families of holomorphic maps, such as quadratic rational maps, cubic polynomials, or other families in which critical orbit relations reduced the space to simple families of maps: for instance, families of polynomials of degrees d ≥ 2 with a single critical point of higher multiplicity. Only in recent years has the progress achieved so far allowed people to shift interest towards higher-dimensional families of iterated maps, such as general polynomials of degree d ≥ 2. The study of iterated rational maps, as opposed to polynomials, seems much more difficult, mainly because of lack of a good partition to obtain a good encoding for symbolic dynamics: the superattracting fixed point at ∞, and the dynamic rays emanating from it, are important ingredients for deep studies of polynomials that are not available for general rational maps. A notable exception are rational maps that arise from Newton maps of polynomials: for these, it seems that good partitions for symbolic dynamics are indeed possible. Transcendental iteration theory has been much less visible for a long time, even though its study goes back to Fatou (we will even treat a question of Euler in Section 4), and a solid body of knowledge has been developed by Baker and coauthors, and later also by Eremenko and Lyubich and by Devaney and coauthors, for more than four decades. Complex dynamics is known for employing methods from many different fields of mathematics, including geometry, complex analysis, algebra and even number theory. Transcendental dynamics unites two different general directions of research: there is a substantial body of knowledge coming from value distribution theory that often yields very general results on large classes of iterated transcendental functions; among the key contributors to this direction of research are Baker, Bergweiler, Eremenko, Rippon, and Stallard. The other direction of research sees transcendental functions as limits of rational functions and employs methods adapted from polynomial or rational iteration; here one usually obtains results on more specific maps or families of maps, most often the prototypical families of exponential or cosine maps; this direction of research was initiated by Devaney and coauthors. Others, like Lyubich and Rempe, have worked from both points of view. In recent years, transcendental iteration theory has substantially gained interest. There have recently been international conferences specifically on transcendental iteration theory, and at more general conferences transcendental dynamics is obtaining more visibility. In this survey article, we try to introduce the reader several aspects of transcendental dynamics. It is based on lecture notes of the CIME summer school held in Cosenza/Italy in summer 2008, but substantially expanded. The topic of that course, “dynamics of entire functions”, also became the title of this article. We thus focus almost entirely on entire functions: their dynamical theory is much simpler than the theory of general meromorphic transcendental functions, much in the same way that polynomial iteration theory is much simpler (and more successful) than rational iteration theory. However, we believe that some of the successes of the polynomial Dynamics of Entire Functions 297 theory still await to be carried over to the world of entire functions, and that some of the key tools (such as dynamic rays) are currently being developed. In this survey article, we try to relate the two points of view on entire functions: that on large classes of entire functions and that on specific prototypical families of entire maps: the simplest families of maps are the exponential family z 7→ λez with a single asymptotic value, and the cosine family z 7→ aez + be−z with two critical values. We try to cover several of the key topics in the theory of entire dynamics. This article is written for readers with a solid background in one-dimensional complex analysis, and a nodding acquaintance of complex dynamics of polynomials. Much more than what we need is provided in Milnor’s now-classical book [Mi06]. In Section 1, we introduce the basic concepts of complex dynamics, including the Fatou and Julia sets, and more specifically the basic concepts of transcendental dynamics such as singular and asymptotic values. We introduce the important set I( f ) of escaping points, review some basic local fixed point theory, and describe the important Zalcman lemma with applications. Section 2 discusses the possibilities for the Fatou set of entire functions and especially highlight those features that are not known from the rational or polynomial theory: Baker domains, wandering domains, and “Baker wandering domains”. The space of general entire functions is a huge space, and many results require the restriction to smaller families of maps: sometimes because tools are lacking to prove results in greater generality, but sometimes also because the space of entire functions is so big that many different dynamical properties are possible, and satisfactorily strong results are possible only when restricting to maps with specific properties. Section 3 introduces important classes of entire maps that are often useful, especially the Eremenko-Lyubich class B of entire functions of bounded type and the class S of functions of finite type. Section 4 is an overview on results on the set I( f ) of escaping points: just as for polynomials, these points often have useful properties that are comparatively easy to investigate, and thus make it possible to establish interesting properties of the Julia set (and sometimes the Fatou set). Section 5 discusses a number of properties on the Hausdorff dimension of Julia set; this is a rich and active area with a number of interesting and sometimes surprising results. Section 6 is a brief introduction to parameter spaces of entire functions. We briefly describe a general result on natural parameter spaces of entire functions, and then discuss exponential parameter space as the best-studied parameter space and prototypical parameter space of entire functions, in analogy to the Mandelbrot set for quadratic polynomials. Section 7 is not directly concernedwith the dynamics of entire functions, but with Newton methods of entire functions: these are special meromorphic functions; we hope that, just as in the rational case, these will be meromorphic functions that can be investigated relatively easily; we describe a number of known results on them. In Section 8, we list a small number of questions that remain open in the field: some of them have remained open for a long time, while others are new. A research area remains lively as long as it still has open questions. 298 Dierk Schleicher In a brief appendix, we state a few important theorems from complex analysis that we use throughout. The research field of complex dynamics is large and active, and many people are working on it from many different points of view. We have selected some topics that we find particularly interesting, and acknowledge that there are a number of other active and interesting topics that deserve no less attention. We mention in particular results on measure theory, including the thermodynamic formalism (see Urbański [Ur03] for a recent survey). Further important omitted areas that we should mention are Siegel disks and their boundaries (see e.g., Rempe [Re04]), questions of linearizations and small cycles (see e.g. Geyer [Ge01]), the construction of entire maps with specific geometric or dynamic properties (here various results of Bergweiler and Eremenko should be mentioned), the relation of transcendental dynamics to Nevanlinna theory, and Thurston theory for transcendental maps (see Selinger [Se09]). We tried to give many references to the literature, but are acutely aware that the literature is vast, and we apologize to those whose work we failed to mention. Many further references can be found in the 1993 survey article of Bergweiler [Be93] on the dynamics of meromorphic functions. The illustration on the first page shows the Julia set of a hyperbolic exponential map with an attracting periodic point of period 26. The Fatou set is in white. We thank Lasse Rempe for having provided this picture. ACKNOWLEDGEMENTS. I would like to thank my friends and colleagues for many interesting and helpful discussions we have had on the field, and in particular on drafts of this manuscript. I would like to especially mention Walter Bergweiler,Jan Cannizzo, Yauhen Mikulich, Lasse Rempe, Phil Rippon, Nikita Selinger, and Gwyneth Stallard. And of course I would like to thank the CIME foundation for having made possible the summer school in Cosenza, and Graziano Gentili and Giorgio Patrizio for having made this such a memorable event! 1 Fatou and Julia Set of Entire Functions Throughout this text, f will always denote a transcendental entire function f : C→C. The dynamics is to a large extent determined by the singular values, so we start with their definition. Definition 1.1 (Singular Value). A critical value is a point w = f (z) with f 0(z) = 0; the point z is a critical point. An asymptotic value is a point w ∈ C such that there exists a curve γ : [0,∞)→ C so that γ(t)→ ∞ and f (γ(t))→ w as t → ∞. The set of singular values of f is the closed set S( f ) := {critical and asymptotic values} . This definition is not completely standard: some authors do not take the closure in this definition. Dynamics of Entire Functions 299 A postsingular point is a point on the orbit of a singular value. In any dynamical system, it is useful to decompose the phase space (in this case, the dynamical plane C) into invariant subsets. In our case, we will mainly consider the Julia set J( f ), the Fatou set F( f ), and the escaping set I( f ), as well as certain subsets thereof. Definition 1.2 (Fatou and Julia Sets). The Fatou set F( f ) is the set of all z ∈ C that have a neighborhoodU on which the family of iterates f ◦n forms a normal family (in the sense of Montel; see Definition A.1). The Julia set J( f ) is the complement of the Fatou set: J( f ) := C\F( f ). A connected component of the Fatou set is called a Fatou component. Definition 1.3 (The Escaping Set). The set I( f ) is the set of points z ∈ C with f ◦n(z)→ ∞. Clearly, F( f ) is open and J( f ) is closed, while in general, I( f ) is neither open nor closed. All three sets J( f ), F( f ), and I( f ) are forward invariant, i.e., f (F( f ))⊂ F( f ) etc.; this is true by definition. Note that equality may fail when f has omitted values: for instance for z 7→ 0.1ez, the Fatou set contains a neighborhood of the origin, but 0 is an omitted value. It is easy to see that for each n≥ 1, F( f ◦n) = F( f ) and hence J( f ◦n) = J( f ), and of course I( f ◦n) = I( f ). Theorem 1.4 (The Julia Set). The Julia set is non-empty and unbounded and has no isolated points. The Fatou set may or may not be empty. An example of an entire function with empty Fatou set is z 7→ ez [Mis81]. More generally, the Fatou set is empty for every entire function of finite type (see Definition 3.1) for which all singular values are either preperiodic or escape to ∞ (Corollary 3.14). We will describe the Fatou set, and give examples of different types of Fatou components, in Section 2. The Fatou set is non-empty for instance for any entire function with an attracting periodic point. The fact that J( f ) is non-emptywas established by Fatou in 1926 [Fa26]. We give here a simple proof due to Bargmann [Ba99]. We start with a preparatory lemma. Lemma 1.5 (Existence of Periodic Points). Every entire function, other than a polynomial of degree 0 or 1, has at least two periodic points of period 1 or 2. Remark 1.6. This is a rather weak result with a simple proof; a much stronger result is given, without proof, in Theorem 1.21. Proof. Consider an entire function f and define a meromorphic function via g(z) := ( f ◦ f (z)− z)/( f (z)− z). Suppose that g is constant, say g(z) = c for all z ∈ C, hence f ◦ f (z)− z = c( f (z)− z). If c = 0, then f ◦ f = id, so f is injective and thus a polynomial of degree 1. If c= 1, then f ◦ f = f , so for each z ∈ C, the value f (z) is a fixed point of f : this implies that either f = id and every z ∈ C is a fixed point, or fixed points 300 Dierk Schleicher of f are discrete and thus f is constant. If c 6∈ {0,1}, then differentiation yields ( f 0 ◦ f ) f 0 −1= c f 0 −c or f 0( f 0 ◦ f −c) = 1−c. Since c 6= 1, it follows that f 0 omits the value 0 and f 0 ◦ f omits the value c 6= 0, so f 0 can assume the value c only at the omitted values of f . By Picard’s Theorem A.4, it follows that f 0 is constant and thus f is a polynomial of degree at most 1. In our case, f is not a polynomial of degree 0 or 1 by hypothesis, so g is a non-constant meromorphic function. Suppose p ∈ C is such that g(p) ∈ {0,1,∞}. If g(p) = ∞, then f (p) = p; if g(p) = 0, then f ( f (p)) = p; and if g(p) = 1, then f ( f (p)) = f (p). If g is transcendental, then by Picard’s theorem there are infinitely many p ∈ C with g(p) ∈ {0,1,∞}. If g is a non-constant rational map (which in fact never happens), then there are p0, p1, p∞ ∈ C with g(pi) = i, and at least two of them are in C. ut As an example, the map f (z) = ez + z has no fixed points; in this case g(z) = ee + 1 has no p ∈ C with g(p) ∈ {1,∞}, but of course infinitely many p with g(p) = 0 (corresponding to periodic points of period 2). Proof of Theorem 1.4. Let f be an entire function other than a polynomial of degree 0 or 1. By Lemma 1.5, f has (at least) two periodic points of period 1 or 2; replacing f by f ◦ f if necessary, we may suppose that f has two fixed points, say at p, p0 ∈C (note that J( f ) = J( f ◦ f )). If | f 0(p)| > 1, then p ∈ J( f ). In order to show that J( f ) 6= / 0, we may assume that p ∈ F( f ) and in particular that | f 0(p)|≤ 1. LetW be the Fatou component containing p. If | f 0(p)| = 1, then any limit function of the family of iterates { f ◦n|W} is nonconstant. So let f ◦n j be a subsequence that converges to a non-constant limit function; then f ◦(n j+1−n j) converges to the identity on W , and this implies that f |W is injective. IfW = C, then this is a contradiction to the choice of f , hence F( f ) 6= C. The final case we have to consider is | f 0(p)| < 1; for convenience, we may suppose that p = 0. If F( f ) = C, then f ◦n → 0 uniformly on compacts in C. We will show that f is a polynomial of degree at most 1. Let D be an open disk centered at p such that f (D) ⊂ D. For N ∈ N, let Dn := f ◦(−n)(D); since D is simply connected, it follows that also Dn is simply connected. Let rn be maximal so that the round circle ∂Drn(0)⊂ Dn; this implies that Drn ⊂ Dn. We have Dn ⊂ Dn+1 and S nDn = C, hence rn+1 ≥ rn and rn→ ∞. As usual, for r > 0, define M(r; f ) := max{| f (z)| : |z| = r}. For each n ∈ N, define maps hn : D → C via hn(z) = M(rn/2; f )−1 f (rnz). All hn satisfy hn(0) = 0 andM(1/2;hn) = 1. Let cn := sup{r > 0: ∂Dr(0)⊂ hn(D)} . If some cn satisfies cn < 1, then there are points an,bn ∈C with |an| = 1, |bn| = 2 so that hn(D)∩{an,bn} = / 0. If this happens for infinitely many n, then we can extract a subsequence with cn < 1, and it follows easily from Montel’s theorem that the hn form a normal family. After extracting another subsequence, we may suppose that the hn converge to a holomorphic limit function h : D→ C that inherits from the hn the properties that h(0) = 0 andM(1/2;h) = 1. Thus h is non-constant and there is Dynamics of Entire Functions 301 an r > 0 with Dr(0)⊂ h(D). But then cn > r/2 for all large n. This implies that no subsequence of the cn tends to 0, and hence that inf{cn} > 0. There is thus a c> 0, and there are cn > c with ∂Dc0n(0)⊂ hn(D) for all large n. This implies ∂Dc0nM(rn/2; f )(0)⊂ f (Drn(0))⊂ f (Dn)⊂ Dn and thus, by the definition of rn,