Table Of ContentJohn 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
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Wiesbaden, 2000
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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
