John Milnor Dynamics in One Complex Variable Introductory Lectures Second Edition John Milnor Dynamics in One Complex Variable John Milnor Dynamics in One Complex Variable Introductory Lectures 2nd Edition I I v1eweg ]ohnMilnor Institute for Mathematical Sciences State University of New York at Stony Brook Stony Brook, NY 11794-3651 USA 1 st Edition 1999 2nd Edition 2000 AII rights reserved © Springer Fachmedien Wiesbaden 2000 Urspriinglich erschienen bei Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/ Wiesbaden, 2000 No part of this publication may be reproduced, stored in a retrieval system or transmitted, mechanical, photocopying or otherwise without prior per mission of the copyright holder. www.vieweg.de Cover design: Ulrike Weigel, www.CorporateDesignGroup.de ISBN 978-3-528-13130-2 ISBN 978-3-663-08092-3 (eBook) DOI 10.1007/978-3-663-08092-3 v TABLE OF CONTENTS Preface . . . . . vi Chronological Table vii Riemann Surfaces 1. Simply Connected Surfaces . . . . . . . . 1 2. Universal Coverings and the Poincare Metric 13 3. Normal Families: Mantel's Theorem . . . . 29 Iterated Holomorphic Maps 4. Fatou and Julia: Dynamics on the Riem~ Sphere 38 5. Dynamics on Hyperbolic Surfaces 54 6. Dynamics on Euclidean Surfaces . 63 7. Smooth Julia Sets . . . . . . . 67 Local Fixed Point Theory 8. Geometrically Attracting or Repelling Fixed Points 73 9. Bottcher's Theorem and Polynomial Dynamics 86 10. Parabolic Fixed Points: the Leau-Fatou Flower 99 11. Cremer Points and Siegel Disks . . . . . . . 116 Periodic Points: Global Theory 12. The Holomorphic Fixed Point Formula for Rational Maps 132 13. Most Periodic Orbits Repel . 140 14. Repelling Cycles are Dense in J . . . . . . . . . . . 143 Structure of the Fatou Set 15. Herman Rings. . . . . . . . . . . . . . . . 148 16. The Sullivan Classification of Fatou Components 153 Using the Fatou Set to study the Julia Set 17. Prime Ends and Local Connectivity . 160 18. Polynomial Dynamics: External Rays 173 19. Hyperbolic and Subhyperbolic Maps . 189 Appendix A. Theorems from Classical Analysis 202 Appendix B. Length-Area-Modulus Inequalities 208 Appendix C. Rotations, Continued Fractions, and Rational Approximation 216 Appendix D. Remarks on Two Complex Variables . 226 Appendix E. Branched Coverings and Orbifolds . 229 Appendix F. No Wandering Fatou Components 234 Appendix G. Parameter Space . . . . . . . 240 Appendix H. Remarks on Computer Graphics 243 References 246 Index . . . . . . . . . . . . . . . . . . 256 vi PREFACE These notes will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on introductory lectures given at Stony Brook during the Fall Term of 1989-90 and also in later years. I am grateful to the audiences for a great deal of constructive criticism, and to Branner, Douady, Hubbard, and Shishikura who taught me most of what I know in this field. Also, I want to thank A. Poirier, S. Zakeri, and R. Perez for their extremely helpful criticisms of various drafts. There have been a number of extremely useful surveys of holomorphic dynamics over the years - those of Brolin, Douady, Blanchard, Lyubich, Devaney, Keen, and Eremenko-Lyubich, as well as the textbooks by Bear don, Steinmetz, and Carleson-Gamelin, are particularly recommended to the reader. (Compare the list of references at the end, and see Alexander for historical information.) This subject is large and rapidly growing. These lectures are intended to introduce the reader to some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. The necessary material can be found for example in Ahlfors 1966, Hocking and Young, Munkres, and vVillmore. However, two big theorems will be used here without proof, namely the Uniformization Theorem in §1 and the existence of solutions for the measurable Beltrami equation in Appendix F. (See the references in those sections.) John Milnor, Stony Brook, June 1999 vii CHRONOLOGICAL TABLE Following is a list of some of the founders of the field of complex dy namics. Ernst Schroder 1841-1902 Hermann Amandus Schwarz 1843-1921 Henri Poincare 1854-1912 Gabriel Kcenigs 1858-1931 Leopold Leau 1868-1940(?) Lucjan Emil Bottcher 1872- ? Samuel Lattes 1873-1918 Constantin Caratheodory 1873-1950 Paul Montel 1876-1975 Pierre Fatou 1878-1929 Paul Koebe 1882-1945 Arnaud Denjoy 1884-1974 Gaston Julia 1893-1978 Carl Ludwig Siegel 1896-1981 Hubert Cremer 1897-1983 Herbert Grotzsch 1902-1993 Charles Morrey 1907-1984 Lars Ahlfors 1907-1996 L~pman Bers 1914-1993 Among the many present day workers in the field, let me mention a few whose work is emphasized in these notes: I. Noel Baker (1932), Adrien Douady (1935), Dennis P. Sullivan (1941), Michael R. Herman (1942), Bodil Branner (1943), John Hamal Hubbard (1945), William P. Thurston (1946), Mary Rees (1953), Jean-Christophe Yoccoz (1955), Curtis Mc Mullen (1958), Mikhail Y. Lyubich (1959), and Mitsuhiro Shishikura (1960). RIEMANN SURFACES §1. Simply Connected Surfaces The first three sections will present an overview of some background material. If V c C is an open set of complex numbers, a function f : V ---+ C is called holomorphic (or "complex analytic") if the first derivative z I-t f'(z) = lim (f(z +h)- f(z))/h h---+0 is defined and continuous as a function from V to C , or equivalently if f has a power series expansion about any point zo E V which converges to f in some neighborhood of zo . (See for example Ahlfors 1966.) Such a function is conformal if the derivative f' (z) never vanishes, and univalent if it is conformal and one-to-one. (In particular, our conformal maps must always preserve orientation.) By a Riemann surface S we mean a connected complex analytic man ifold of complex dimension one. Thus S is a connected Hausdorff space. Furthermore, in some neighborhood U of an arbitrary point of S we can choose a local uniformizing parameter (or "coordinate chart") which maps U homeomorphically onto an open subset of the complex plane C , with the following property: In the overlap U n U' between two such neighbor hoods, each of these local uniformizing parameters can be expressed as a holomorphic function of the other. By definition, two Riemann surfaces S and S' are conformally isomor phic (or biholomorphic) if and only if there is a homeomorphism from S 2 RIEMANN SURFACES onto S' which is holomorphic, in terms of the respective local uniformizing parameters. (It is an easy exercise to show that the inverse map S' --+ S must -then also be holomorphic.) In the special case S = S' , such a con formal isomorphism S --+ S is called a conformal automorphism of S . Although there is an uncountable infinity of conformally distinct Rie mann surfaces, the classification is very easy to describe in the simply con nected case. (By definition, the surface S is simply connected if every map from a circle into S can be continuously deformed to a constant map. Com pare §2.) According to Poincare and Koebe, there are only three simply connected Riemann surfaces, up to isomorphism: 1.1. Uniformization Theorem. Any simply connected Rie mann surface is conformally isomorphic either (a) to the plane C consisting of all complex numbers z =X +iy, (b) to the open disk lD> C C consisting of all z with lzl2 = x2 + y2 < 1 ' or (c) to the Riemann sphere C consisting of C together with a 1/ point at infinity , using ( = z as local uniformizing param eter in a neighborhood of the point at infinity. This is a generalization of the classical Riemann Mapping Theorem. We will refer to these three cases as the Euclidean, hyperbolic, and spherical cases respectively. (Compare §2.) I will not try to give a proof of 1.1. The proof, which is quite difficult, may be found in Springer, or Farkas & Kra, or Ahlfors (1973), or Nevanlinna, or in Beardon (1984). (See also Fisher, Hubbard and Wittner.) By assuming this result, we will be able to get more quickly to interesting ideas in holomorphic dynamics. D The Open Disk lD> • For the rest of this section, we will discuss these three surfaces in more detail. We begin with a study of the unit disk lD> . 1.2. Schwarz Lemma. If f : lD> --+ ][)) is a holomorphic map with f(O) = 0, then the derivative at the origin satisfies lf'(O)I:::; 1. If equality holds, lf'(O)I = 1, then f is a rotation = = about the origin. That is, f ( z) cz for some constant c f' ( 0) on the unit circle. On the other hand, if If' ( 0) I < 1 , then lf(z) I < lzl for all z =/= 0. (The Schwarz Lemma was first proved, in this generality, by Caratheo dory.) I Remarks. If If' (0) = 1 , it follows that f is a conformal automor- 1. SIMPLY CONNECTED SURFACES 3 phism of the unit disk. But if lf'(O)I < 1 then f cannot be a conformal automorphism of ID>, since the composition with any g : (ID>, 0) ~ (ID>, 0) would have derivative g'(O)f'(O) ¥ 1 . The example f(z) = z2 shows that f may map ID> onto itself even when lf(z)l < lzl for all z =/= 0 in ID>. Proof of 1.2. We use the Maximum Modulus Principle, which asserts that a non-constant holomorphic function cannot attain its maximum abso lute value at any interior point of its region of definition. First note that the quotient function q(z) = f(z)/z is well defined and holomorphic through out the disk ID> (as one sees by dividing the local power series for f by z ) . Since lq(z)l < 1/r when lzl = r < 1, it follows by the Maximum Modulus Principle that lq(z)l < 1/r for all z in the disk lzl ::; r. Since this is I I ::; true for all r ~ 1 , it follows that q ( z) 1 for all z E ID> • Again by the Maximum Modulus Principle, we see that the case lq(z)l = 1, for some z in the open disk, can occur only if the function q(z) is constant. If we = exclude this case f(z)/z c, then it follows that lq(z)l = lf(z)/zl < 1 for all z =/= 0 , and similarly that lq(O) I = If' (0) I < 1 . 0 Here is a useful variant statement. 1.2'. Cauchy Derivative Estimate. If f maps the disk of radius r about zo into some disk of radius s , then lf'(zo)l ::; s/r . Proof. This follows easily from the Cauchy integral formula (see for example Ahlfors): Set g(z) = f(z + zo) +constant, so that g maps the disk ID>r centered at the origin to the disk ID>s centered at the origin. Then f'(zo) = g'(O) = ~ 1 g(z) dz 27rz Jlzi=r z2 1 for all r1 < r , and the conclusion follows. 0 (An alternative proof, based on the Schwarz Lemma, is described in Problem 1-a below. With an extra factor of 2 on the right, this inequality would follow immediately from 1.2 simply by linear changes of variable, since the target disk of radius s must be contained in the disk of radius 2s centered at the image f (z o) .) As an easy corollary, we obtain the following. 1.3. Theorem of Liouville. A bounded function f which is defined and holomorphic everywhere on C must be constant. For in this case we have s fixed but r arbitrarily large, hence f' must be identically zero. o As another corollary, we see that our three model surfaces really are

Dynamics in One Complex Variable: Introductory Lectures

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