The arithmetic of dynamical systems
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Graduate Texts
in Mathematics
Joseph H. Silverman
The Arithmetic of
Dynamical Systems
€1 Springer
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Graduate Texts in Mathematics
241
Editorial Board
S. Axler K.A. Ribet
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TAKEun/ZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
OxTOBY. Measure and Category. 2nd ed.
ScHAEFER. Topological Vector Spaces.
2nd ed.
HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
MAc LANE. Categories for the Working
Mathematician. 2nd ed.
Hu GHEs/PIPER. Projective Planes.
J.-P. SERRE. A Course in Arithmetic.
TAKEun/ZARING. Axiomatic Set Theory.
HuMPHREYS. Introduction to Lie Algebras and Representation Theory.
CoHEN. A Course in Simple Homotopy
Theory.
CoNWAY. Functions of One Complex
Variable I. 2nd ed.
BEALS. Advanced Mathematical
Analysis.
ANDERSON/FULLER. Rings and
Categories of Modules. 2nd ed.
GoLUBITSKY/GuiLLEMIN. Stable
Mappings and Their Singularities.
BERBERIAN. Lectures in Functional
Analysis and Operator Theory.
WINTER. The Structure o f Fields.
RosENBLATT. Random Processes. 2nd ed.
HALMOS. Measure Theory.
HALMOS. A Hilbert Space Problem
Book. 2nd ed.
HusEMOLLER. Fibre Bundles. 3rd ed.
HuMPHREYS. Linear Algebraic Groups.
BARNEs/MAcK. An Algebraic
Introduction to Mathematical Logic.
GREUB. Linear Algebra. 4th ed.
HoLMES. Geometric Functional
Analysis and Its Applications.
HEWITT/STROMBERG. Real and Abstract
Analysis.
MANES. Algebraic Theories.
KELLEY. General Topology.
ZARISKI/SAMUEL. Commutative
Algebra. Voi.I.
ZARISKIISAMUEL. Commutative
Algebra. Voi.II.
JACOBSON. Lectures in Abstract Algebra
I. Basic Concepts.
JACOBSON. Lectures i n Abstract Algebra
II. Linear Algebra.
JACOBSON. Lectures in Abstract Algebra
III. Theory of Fields and Galois
Theory.
HIRSCH. Differential Topology.
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SPITZER. Principles of Random Walk.
2nd ed.
ALEXANDERIWERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
KELLEYINAMIOKA et al. Linear
Topological Spaces.
MoNK.Mathematical Logic.
GRAUERT/FRITZSCHE. Several Complex
Variables.
ARVESON. An Invitation to C*-Algebras.
KEMENYISNELLIKNAPP. Denumerable
Markov Chains. 2nd ed.
APOSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
J.-P. SERRE. Linear Representations of
Finite Groups.
GILLMAN/JERISON. Rings of
Continuous Functions.
KENDIG. Elementary Algebraic
Geometry.
LoE:vE. Probability Theory I. 4th ed.
LoE:vE. Probability Theory II. 4th ed.
MmsE. Geometric Topology in
Dimensions 2 and 3.
SAcHs/Wu. General Relativity for
Mathematicians.
GRUENBERG/WEIR. Linear Geometry.
2nd ed.
EDWARDS. Fermat's Last Theorem.
KLIN GENBERG. A Course in Differential
Geometry.
HARTSHORNE. Algebraic Geometry.
MANIN. A Course in Mathematical
Logic.
GRAVERIWATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
BRowN/PEARCY. Introduction to
Operator Theory I: Elements of
Functional Analysis.
MASSEY. Algebraic Topology: An
Introduction.
CROWELL/Fox. Introduction to Knot
Theory.
KoBLITZ. p-adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
LANG. Cyclotomic Fields.
ARNOLD.Mathematical Methods in
Classical Mechanics. 2nd ed.
WHITEHEAD. Elements o f Homotopy
Theory.
KARGAPOLOV/MERLZIAKOV.
Fundamentals of the Theory of Groups.
BoLLOBAS. Graph Theory.
(continued after index)
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Joseph H. Silverman
The Arithmetic of
Dynamical Systems
With 11 Illustrations
�Springer
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Joseph H. Silverman
Department of Mathematics
Brown University
Providence, RI 02912
USA
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA
K.A. Ribet
Mathematics Department
University of California, Berkeley
Berkeley, CA 94720-384
USA
Mathematics Subject Classification (2000): 11- 01, 1 1G99, 14G99, 37-01, 37Fl 0
ISBN-13: 978-0-387-69903-5
e-ISBN-13: 978-0-387-69904-2
Library of Congress Control Number: 2007923502
Printed on acid-free paper.
© 2007 Springer Science+ Business Media, LLC
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Preface
This book is designed to provide a path for the reader into an amalgamation of two
venerable areas of mathematics, Dynamical Systems and Number Theory. Many of
the motivating theorems and conjectures in the new subject of Arithmetic Dynamics
may be viewed as the transposition of classical results in the theory of Diophantine
equations to the setting of discrete dynamical systems, especially to the iteration
theory of maps on the projective line and other algebraic varieties. Although there is
no precise dictionary connecting the two areas, the reader will gain a flavor of the
correspondence from the following associations:
Diophantine Equations
Dynamical Systems
rational and integral
points on varieties
rational and integral
points in orbits
torsion points on
abelian varieties
periodic and preperiodic
points of rational maps
There are a variety of topics covered in this volume, but inevitably the choice
reflects the author's tastes and interests. Many related areas that also fall under the
heading of arithmetic or algebraic dynamics have been omitted in order to keep the
book to a manageable length. A brief list of some of these omitted topics may be
found in the introduction.
Online Resources
The reader will find additonal material, references and errata at
/>
Acknowledgments
The author has consulted a great many sources in writing this book. Every attempt
has been made to give proper attribution for all but the most standard results. Much
of the presentation is based on courses taught at Brown University in 2000 and 2004,
and the exposition benefits greatly from the comments of the students in those
v
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vi
Preface
courses. In addition, the author would like to thank the many students and math
ematicians who read drafts and/or offered suggestions and corrections, including
Matt Baker, Rob Benedetto, Paul Blanchard, Rex Cheung, Bob Devaney, Graham
Everest, Liang-Chung Hsia, Rafe Jones, Daniel Katz, Shu Kawaguchi, Michelle
Manes, Patrick Morton, Curt McMullen, Hee Oh, Giovanni Panti, Lucien Szpiro,
Tom Tucker, Claude Viallet, Tom Ward, Xinyi Yuan, Shou-Wu Zhang. An especial
thanks is due to Matt Baker, Rob Benedetto and Liang-Chung Hsia for their help in
navigating the treachorous shoals of p-adic dynamics. The author would also like to
express his appreciation to John Milnor for a spellbinding survey talk on dynamical
systems at Union College in the mid-1980s that provided the initial spark leading
eventually to the present volume. Finally, the author thanks his wife, Susan, for her
support and patience during the many hours occupied in writing this book.
Joseph H. Silverman
January 1, 2007
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Contents
v
Preface
Introduction
1
Exercises
7
1 An Introduction to Classical Dynamics
1.1
1 .2
1.3
1 .4
1 .5
1 .6
2
Rational Maps and the Projective Line . . . . . . .
Critical Points and the Riemann-Hurwitz Formula .
Periodic Points and Multipliers
The Julia Set and the Fatou Set . . . . . . . . . . .
Properties of Periodic Points . . . . . . . . . . . .
Dynamical Systems Associated to Algebraic Groups
Exercises . . . . . . . . . . . . . . . . . . . . . . .
Dynamics over Local Fields: Good Reduction
2.1
2.2
2.3
2.4
2.5
2.6
2. 7
The Nonarchimedean Chordal Metric . .
Periodic Points and Their Properties . . .
Reduction of Points and Maps Modulo p .
The Resultant of a Rational Map . . .
Rational Maps with Good Reduction .
Periodic Points and Good Reduction .
Periodic Points and Dynamical Units .
Exercises . . . . . . . . . . . . . . .
3 Dynamics over Global Fields
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Height Functions . . . . . . . . . . . .
Height Functions and Geometry . . . .
The Uniform Boundedness Conjecture .
Canonical Heights and Dynamical Systems
Local Canonical Heights . .
Diophantine Approximation . . . . . . .
Integral Points in Orbits . . . . . . . . . .
Integrality Estimates for Points in Orbits .
Periodic Points and Galois Groups
vii
9
9
12
18
22
27
28
35
43
43
47
48
53
58
62
69
74
81
81
89
95
97
102
104
108
1 12
122
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viii
Contents
3.10 Equidistribution and Preperiodic Points
3. 1 1 Ramification and Units in Dynatomic Fields
Exercises . . . . . . . . .
4 Families of Dynamical Systems
4. 1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.1 0
4.1 1
5
6
126
129
135
147
Dynatomic Polynomials . .
Quadratic Polynomials and Dynatomic Modular Curves .
The Space Ratd of Rational Functions . . . . . .
The Moduli Space Md of Dynamical Systems . . . . . .
Periodic Points, Multipliers, and Multiplier Spectra . . .
The Moduli Space M2 of Dynamical Systems of Degree 2
Automorphisms and Twists .
General Theory of Twists . . . . . . . . . .
Twists of Rational Maps . . . . . . . . . .
Fields of Definition and the Field of Moduli
Minimal Resultants and Minimal Models
Exercises . . . . . . . . . . . . . . . . .
.
148
155
168
174
179
188
195
199
203
206
218
224
Dynamics over Local Fields: Bad Reduction
239
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
240
242
248
254
257
263
268
276
287
294
3 12
Absolute Values and Completions . . . .
A Primer on Nonarchimedean Analysis . . . . . . . . .
Newton Polygons and the Maximum Modulus Principle .
The Nonarchimedean Julia and Fatou Sets
The Dynamics of (z2 - z) jp . . . . .
A Nonarchimedean Montel Theorem .
Periodic Points and the Julia Set . . .
Nonarchimedean Wandering Domains
Green Functions and Local Heights
Dynamics on Berkovich Space
Exercises . . . . . . . . . . . . . .
Dynamics Associated to Algebraic Groups
6. 1
6.2
6.3
6.4
6.5
6.6
6. 7
6.8
Power Maps and the Multiplicative Group
Chebyshev Polynomials . . . . . .
A Primer on Elliptic Curves . . .
General Properties of Lattes Maps
Flexible Lattes Maps . . . . . . .
Rigid Lattes Maps . . . . . . . . .
Uniform Bounds for Lattes Maps .
Affine Morphisms and Commuting Families .
Exercises . . . . . . . . . . . . . . . . . . .
325
325
328
336
350
355
364
368
375
380
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Contents
7 Dynamics in Dimension Greater Than One
7.1
7.2
7.3
7.4
Dynamics of Rational Maps on Projective Space .
Primer on Algebraic Geometry . . . . . . . . . .
The Weil Height Machine . . . . . . . . . . . . .
Dynamics on Surfaces with Noncommuting Involutions .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
ix
387
388
402
407
410
427
Notes on Exercises
441
List of Notation
445
References
451
Index
473
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Introduction
A (discrete) dynamical system consists of a set S and a function ¢ : S
the set S to itself. This self-mapping permits iteration
q;n
=
¢ ¢
o
o . . . o
-+
S mapping
¢ = n1h iterate of ¢.
'-----v-----'
n times
(By convention, ¢0 denotes the identity map on S.)
For a given point E S, the (forward) orbit ofo: is the set
o:
=
=
Oq,(o:) O(o:) {¢n (o:) : 2: 0}.
The point o: is periodic if q;n (o:) o: for some 2: 1. The smallest such is called
the exact period of o:. The point o: is preperiodic if some iterate q;m ( o:) is periodic.
The sets of periodic and preperiodic points of¢ in S are denoted respectively by
Per(¢, S) {o: E S : q;n (o:) o: for some 2: 1 } ,
PrePer(¢, S) {o: E S : q;m+n (o:) q;m (o:) for some n 2: 1, m 2: 0}
{o: S : Oq,(o:) is finite}.
n
=
=
=
n
=
E
n
n
=
=
We write Per(¢) and PrePer( ¢) when the set S is fixed.
Principal Goal of Dynamics
q, ( o:
Classify the points in the set S according to the behavior of
their orbits 0 o:).
If S is simply a set with no additional structure, then typical problems are to de
scribe the sets of periodic and preperiodic points and to describe the possible periods
of periodic points. Usually, however, the set S has some additional structure and
one attempts to classify the points in S according to the interaction of their orbits
with that structure. There are many types of additional structures that may imposed,
including algebraic, topological, metric, and analytic.
Example 0. 1. (Finite Sets) Let S be a finite set and ¢ : S
S a function. Clearly
every point of S is preperiodic, so we might ask for the number of elements in the
set of periodic points
-+
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2
Introduction
Per(¢,S) = {a E S : ¢n(a) = a for some n � 1}.
An interesting class of sets and maps are finite fields S = lFP and maps
¢ : lFP - ---> lFp given by polynomials ¢(z) E lFp[z]. For example, Fermat's Little The
orem says that
and
which gives two extremes for the set of periodic points. A much harder question is to
fix an integer d � 2 and ask for which primes p there is a polynomial ¢ of degree d
satisfying
Similarly, one might fix a polynomial
E
and ask for which primes p is it true that
In particular, are there
infinitely many such primes?
In a similar, but more general, vein, one can look at a rational function ¢ E
inducing a rational map ¢ : IP1 (JF
IP1 (JF Even more generally, one can ask
similar questions for a morphism ¢ :
of any variety
for
lPN.
example
Example 0.2. (Groups) Let G be a group and let ¢ : G----> G be a homomorphism.
Using the group structure, it is often possible to describe the periodic and preperiodic
points of¢ fairly explicitly. The following proposition describes a simple, but impor
tant, example. In order to state the proposition, we recall that the torsion subgroup of
an abelian group G, denoted by G10rs , is the set of elements of finite order in G,
Per(¢,lFp) = lFP .
¢(z) Z[z]
lFP (z)
VjlFP,
Per(¢, lFP ) = lFp·
P ) V(lFp)P ). V(lFp)
---->
---->
V=
G {a E G am =
tors =
:
e
for some m � 1 },
where e denotes the identity element of G.
Proposition 0.3. Let G be an abelian group, let d
¢: G----> G be the d1h power map
Then
¢(a) = ad.
PrePer(¢, G) G
=
>
2 be an integer, and let
tors ·
Proof The simple nature of the map ¢ allows us to give an explicit formula for its
iterates,
¢n (a) adn .
Now suppose that a E PrePer(¢d , G). This means that ¢m+ n(a) ¢m (a) for
some n � 1 and � 0, so ad "'+n = adm. But G is a group, so we can multiply
by a- dm to get ad m+n _ dm = The assumptions on d, and imply that the
exponent is positive, so a E G
Next suppose that a E G say am = and consider the following sequence
of integers modulo
d, d2, d3, d4, . . . modulo
=
=
m
e.
1ors ·
tors,
m:
n
m,
e,
m.
Since there are only finitely many residues modulo m, eventually the sequence has a
repeated element, say di dJ (mod m) with i > j. Then
e and di
¢j
since
dj (mod m ) .
=
¢i (a) = ad' = adj = (a),
Hence a E PrePer(¢).
am =
=
0
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Introduction
3
Example 0.4. (Topological Spaces) Let S be a topological space and let ¢ : S-+ S
be a continuous map. For a given a E S, one might ask for a description of the
accumulation points of 0 q,(a). For example, a point a is called recurrent if it is an
accumulation point of 0q,(a). In other words, a is recurrent if there is a sequence of
integers n1 < n2 < n3 < · · · such that limi-+oo ¢Pi (a) = a, so either a is periodic,
or it eventually returns arbitrarily close to itself.
Example 0.5. (Metric Spaces) Let (S, p) be a compact metric space. For example, S
could be the unit sphere sitting inside IR3, and p(a, /3) the usual Euclidean distance
from a to j3 in IR3 . The fundamental question in this setting is whether points that
start off close to a given point a continue to remain close to one another under re
peated iteration of ¢. If this is true, we say that ¢ is equicontinuous at a; otherwise
we say that ¢ is chaotic at a. (See Section 1 .4 for the formal definition of equicon
tinuity.) Thus if ¢ is equicontinuous at a, we can approximate
Example 0.6. (Arithmetic Sets) An arithmetic set is a set such as Z or Q or a num
ber field that is of number-theoretic interest, but doesn't have a natural underlying
topology. More precisely, an arithmetic set tends to have a variety of interesting
topologies; for example, Q has the archimedean topology induced by the inclusion
Q C lR and the p-adic topologies induced by the inclusions Q C Qp. In the arith
metic setting, the map ¢ is generally a polynomial or a rational map. Here are some
typical arithmetical-dynamical questions, where we take ¢(z) E Q(z) to be a ratio
nal function of degree d 2: 2 with rational coefficients:
•
•
•
Let a E Q be a rational number. Under what conditions can the orbit Oq,(a)
contain infinitely many integer values? In other words, when can 0q, ( a ) n Z
be an infinite set?
Is the set Per(¢, Q) of rational periodic points finite or infinite? If finite, how
large can it be?
Let a E Per(¢) be a periodic point for ¢. It is clear that a is an algebraic
number. What are the arithmetic properties of the field Q( a), or more generally
of the field generated by all of the periodic points of a given period?
What is in this book: We provide a brief summary of the material that is covered.
1.
An Introduction to Classical Dynamics
2.
Dynamics over Local Fields: Good Reduction
We begin in Chapter 1 with a short self-contained overview, without proofs, of
classical complex dynamics on the projective line.
Chapter 2, which starts our study of arithmetic dynamics, considers rational
maps ¢(z) with coefficients in a local field K, for example, K = Qp. The em
phasis in Chapter 2 is on maps that have "good reduction modulo p." The good
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Introduction
4
reduction property imples that many of the geometric properties of ¢ acting on
the points of K are preserved under reduction modulo p. In particular, the map ¢
is p-adically nonexpanding, and periodic points behave well when reduced mod
ulo p. The remainder of the chapter gives applications exploiting these two key
properties of good reduction.
3.
4.
Dynamics over Global Fields
We move on in Chapter 3 to arithmetic dynamics over global fields such as Q
and its finite extensions. Just as in the study of Diophantine equations over global
fields, the theory of height functions plays a key role, and we develop this the
ory, including the construction of the canonical height associated to a rational
map. We discuss rationality of preperiodic points and formulate a general uniform
boundedness conjecture. Using classical results from the theory of Diophantine
approximation, we describe exactly which rational maps ¢ can have orbits con
taining infinitely many integer points, and we give a more precise result saying
that the numerator and denominator of ¢n (a) grow at approximately the same
rate. We consider the extension fields generated by periodic points and describe
their Galois groups, ramification, and units.
Families ofDynamical Systems
At this point we change our perspective, and rather than studying the dynamics
of a single rational map, we consider families of rational maps and the variation
of their dynamical properties. We construct various kinds of parameter and mod
uli spaces, including the space of quadratic polynomials with a point of exact
period N (which are analogues of the classical modular curves X1(N)), the pa
rameter space Ratd of rational functions of degree d, and the moduli space Md
ofrational functions of degree d modulo the natural conjugation action by 12•
In particular, we prove that M2 is an isomorphism to the affine plane A2 . We
also study twists of rational maps, analogous to the classical theory of twists of
varieties, and the field-of-moduli versus field-of-definition problem.
PG
5.
Dynamics over Local Fields: Bad Reduction
6.
Dynamics Associated t o Algebraic Groups
Chapter 5 returns to arithmetic dynamics over local fields, but now in the case of
"bad reduction." It becomes necessary to work over an algebraically closed field,
so we discuss the field Cp and give a brief introduction to nonarchimedean analy
sis and Newton polygons. Using these tools, we define the nonarchimedean Julia
and Fatou sets and prove a version ofMontel's theorem that is then used to study
periodic points and wandering domains in the nonarchimedean setting. This is fol
lowed by the construction of p-adic Green functions and local canonical heights.
The chapter concludes with a short introduction to dynamics on Berkovich space.
The Berkovich projective line lP'8 is path connected, compact, and Hausdorff, yet
it naturally contains the totally disconnected, non-locally compact, non-Hausdorff
space lP' 1 (Cp).
There is a small collection of rational maps whose dynamics are much easier to
understand than those of a general map. These special rational maps are associ
ated to endomorphisms of algebraic groups. We devote Chapter 6 to the study of
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Introduction
5
these maps. The easiest ones are the power maps Md(z) = zd and the Chebyshev
polynomials Td(z) characterized by Td(2cos0) = 2cos(dB). They are associ
ated to the multiplicative group. More interesting are the Lattes maps attached to
elliptic curves. We give a short description, without proofs, of the theory of ellip
tic curves and then spend the remainder of the chapter discussing dynamical and
arithmetic properties of Lattes maps.
7.
Dynamics in Dimension Greater Than One
With a few exceptions, the results in Chapters 1 -6 all deal with iteration of maps
on the one-dimensional space lP' 1 , i.e., they are dynamics of one variable. In Chap
ter 7 we consider some of the issues that arise in studying dynamics in higher
dimensions. We first study a class of rational maps ¢ : lP'N lP'N that are not
everywhere defined. Even over C, the geometry of dynamics of rational maps is
imperfectly understood. We restrict attention to automorphisms ¢ AN ___. AN
and study height functions and rationality of periodic points for such maps. We
next consider morphisms ¢ X ___. X of varieties other than lP'N. In order to
deal with higher-dimensional dynamics, we use tools from basic algebraic geom
etry and Weil 's height machine, which we describe without proof. We then study
arithmetic dynamics, heights, and periodic points on K3 surfaces admitting two
noncommuting involutions L 1 and L2ã The composition  = L 1 t2 provides an
automorphism ¢ X ___. X whose geometric and arithmetic dynamical properties
are quite interesting.
___.
:
:
o
:
What's missing: A book necessarily reflects the author's interests and tastes, while
space considerations limit the amount of material that can be included. There are
thus many omitted topics that naturally fit into the purview of arithmetic dynamics.
Some of these are active areas of current mathematical research with their own liter
ature, including introductory and advanced textbooks. Others are younger areas that
deserve books of their own. Examples of both kinds include the following, some of
which overlap with one another:
•
•
•
Dynamics over finite fields
This includes general iteration of polynomial and rational maps acting on finite
fields, see for example [4 1 , 42, 1 02, 1 06, 1 09, 1 79, 220, 222, 275, 308, 336,
350, 385, 384, 400, 437, 444], and more specialized topics such as permutation
polynomials [275, Chapter 7] that are fields of study in their own right.
Dynamics over function fields
The study of function fields over finite fields has long provided a parallel theory
to the study of number fields, but inseparability and wild ramification often lead
to striking differences, while function fields of characteristic 0 present their own
arithmetic challenges, e.g., they have infinitely many points of bounded height.
The study of arithmetic dynamics over function fields is in its infancy. For a hand
ful of results, see [20, 6 1 , 85, 1 07, 207, 279, 354, 356, 4 1 5].
Iteration o fpower series
There is an extensive literature, but no textbook, on the iteration properties of
formal, p-adic, and Puiseux power series. Among the fundamental problems are
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Introduction
6
the classification of nontrivial commuting power series (to what extent do they
come from formal groups?) and the description of preperiodic points. See for
example [246,266,267,268,269,270,27 1 ,272,273,274,280,28 1 ,282,283,
284,389,390].
•
•
Algebraic dynamics
There is no firm line between arithmetic dynamics and algebraic dynamics, and
indeed much of the material in this book is quite algebraic. Some topics of an
algebraic nature that we do not cover include irreducibility of iterates [6, 1 7,1 1 3,
1 1 5,345,346,425],formal transformations and algebraic identities [55,80,79],
and various results of an algebro-geometric nature [140,149,1 63,392].
Lie groups and homogeneous spaces, ergodic theory, and entropy
This is a beautiful and much studied area of mathematics in which geometry,
analysis, and algebra interact. There are many results of a global arithmetic nature,
including for example hard problems of Diophantine approximation, as well as
an extensive p-adic theory. For an introduction to some of the main ideas and
theorems in this area, see [47, 247, 304,423],and for other arithmetic aspects
of ergodic theory and entropy, including relations with height functions, ergodic
theory in a nonarchimedean setting, and arithmetic properties of dynamics on
solenoids, see for example [9,31, 1 04, 1 30, 147, 1 60, 1 88, 229, 239, 241 ,25 1 ,
277,278,35 1 ,440,441 ,442,447].
•
Equidistribution in arithmetic dynamics
There are many ways to measure (arithmetic) equidistribution, including via
canonical heights, p-adic measures, and invariant measures on projective and
Berkovich spaces. In Section 3. 1 0 we summarize some basic equidistribution con
jectures and theorems (without proof). For additional material, see [ 1 5,24,28,98,
99,1 68,1 82,209,429,432,450].
•
•
•
•
Topology and arithmetic dynamics on foliated spaces
This surprising connection between these diverse areas of mathematics has been
inverstigated by Derringer in a series of papers [ 1 24,1 25,1 26,1 27,1 28].
Dynamics on Drinfeld modules
It is natural to study local and global arithmetic dynamics in the setting of Drin
feld modules, although only a small amount of work has yet been done. See for
example [1 80,1 8 1 ,1 82,349,395] .
Number-theoretic iteration problems not arising as maps on varieties
A famous example of this type of problem is the notorious 3x + 1 problem,
see [253] for an extensive bibliography. Another problem that people have stud
ied is iteration of arithmetic functions such as Euler's 'P function; see for exam
ple [1 5 1 ,342].
Realizability of integer sequences
(an )
A sequence
of nonnegative integers is said to be realizable if there are a
set S and a function ¢ S ---. S with the property that for all n, the map ¢ has
periodic points of order n. See [1 58] for an overview and [ 1 2,144,1 57,362,363,
4 1 9] for further material on realizable sequences.
:
an
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7
Exercises
Prerequisites: The principal prerequisite for reading this book is basic algebraic
number theory (rings of integers, ideals and ideal class groups, units, valuations and
absolute values, completions, ramification, etc.) as covered, for example, in the first
section of Lang's Algebraic Number Theory [258]. We also assume some knowl
edge of elementary complex analysis as typically covered in an undergraduate course
in the subject. No background in dynamics or algebraic geometry is required; we
summarize and give references as necessary. In particular, to help make the book
reasonably self-contained, we have included introduction/overview material on non
archimedean analysis in Section 5.2, elliptic curves in Section 6.3, and algebraic
geometry in Section 7.2. However, previous familiarity with basic algebraic geome
try will certainly be helpful in reading some parts of the book, especially Chapters 4
and 7.
Cross-references and exercises: Theorems, propositions, examples, etc. are num
bered consecutively within each chapter and cross-references are given in full; for
example, Proposition 3.2 refers to the second labeled item in Chapter 3. Exercises
appear at the end of each chapter and are also numbered consecutively, so Exer
cise 5.7 is the seventh exercise in Chapter 5. There is an extensive bibliography, with
reference numbers in the text given in square brackets.
This book contains a large number of exercises. Some ofthe exercises are marked
with a single asterisk , which indicates a hard problem. Others exercises are marked
with a double asterisk , which means that the author does not know how to solve
them. However, it should be noted that these "unsolved" problems are of varying de
grees of difficulty, and in some cases their designation reflects only the author's lack
of perspicacity. On the other hand, some of the unsolved problems are undoubtedly
quite difficult. The author solicits solutions to the
marked problems, as well as
solutions to the exercises that are posed as questions, for inclusion in later editions.
The reader will find additional notes and references for the exercises on page 441 .
*
**
**
Standard Notation: Throughout this book we use the standard symbols
Z, Q, JR., CC, Fq, Zv, AN , and !P'N
to represent the integers, rational numbers, real numbers, complex numbers, field
with q elements, ring of p-adic integers, N-dimensional affine space, and N-dimen
sional projective space, respectively. Additional notation is defined as it is introduced
in the text. A detailed list of notation may be found on page 445.
Exercises
0.1.
(a)
(b)
(c)
Let S be a set and ¢ : S ---> S a function.
If S is a finite set, prove that ¢ is bijective if and only if Per(¢, S) = S.
In general, prove that if Per(¢, S) = S, then ¢ is bijective.
Give an example of an infinite set S and map ¢ with the property that ¢ is bijective and
Per(¢, S) =/= S.
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8
Exercises
(d) If¢ is injective, prove that PreFer(¢, S) = Per(¢, S).
0.2. Let S be a set, let ¢ : S __... S and 'lj; : S __... S be two maps of S to itself, and suppose
that ¢ and 'lj; commute, i.e., assume that ¢ o 'lj; = 'lj; o ¢.
{a) Prove that'l/J (PrePer(¢) ) C PreFer(¢).
(b) Assume further that 'lj; is a finite-to-one surjective map, i.e., 'lj;(S) = S, and for every x E
S, the inverse image 'lj;- 1 (x ) is finite. Prove that'l/J (PrePer( ¢) ) = PreFer(¢).
(c) We say that a point P E S is an isolatedpreperiodic point of¢ ifthere are integers n >
n
m such that ¢ (P) = ¢m (P) and such that the set
is finite. Suppose that every preperiodic point of ¢ is isolated. Prove that
PreFer(¢)
C
PreFer('lj;) .
Conclude that if the commuting maps ¢ and 'lj; both have isolated preperiodic points,
then PreFer(¢) = PrePer('!j; ).
0.3. Let ¢(z) =zd +a
if gcd(d, p - 1) = 1 .
E
Z[z ] and let p be a prime. Prove that Per(¢, 1Fp) = 1Fp if and only
0.4. Let G be a group and let ¢ : G __... G be a homomorphism.
(a) Prove that Per(¢, G) is a subgroup of G.
(b) Is PreFer(¢, G) a subgroup of G? Either prove that it is a subgroup or give a counterex
ample.
0.5. Let G be a topological group, that is, G is a topological space with a group structure
such that the group composition and inversion laws are continuous maps. Let ¢ : G __... G
be a continuous homomorphism. Exercise 0.4 says that Per(¢, G) is a subgroup of G, so its
topological closure Per(¢, G) is also a subgroup of G. Compute this topological closure for
each of the following examples. (In each example, d ;::::: 2 is a fixed integer.)
(a) G = C* and ¢(a) = a d .
(b) G = JR* and ¢(a) = a d .
(c) G = JRN /ZN and ¢(a) = da mod zN .
+
0.6. (a) Describe Per(¢, Q) for the function ¢(z) = z2 1.
(b) Describe Per(¢, Q) for the function ¢(z) = z2 - 1.
(c) Let ¢(z) E Z[z] be a monic polynomial of degree at least two. Prove that Per(¢, Q) is
finite. (Hint. First prove that Per(¢, Q) c Z.)
(d) Same question as (c), but now ¢(z) E Q[z] has rational coefficients and is not assumed
to be monic.
+
0.7. Let ¢(z) = z 1/z and let a E Q* . Prove that Oq,(a) n Z is finite. What is the largest
number of points that it can contain?
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Chapter 1
An Introduction to Classical
Dynamics
Classically, the subject of discrete complex dynamical systems involves the study of
iteration of polynomial and rational maps on or 1? 1 q. The local theory of analytic
iteration dates back to the 19th century, but the modern theory of global complex
dynamics starts with the foundational works of Fatou [166, 167] and Julia [223] in
1918-1920. From that beginning has arisen a vast literature.
Our goal in this book is to study number-theoretic questions associated to itera
tion of polynomial and rational maps. A standard technique in number theory is to
attempt to answer questions related to a number field K by first studying analogous
questions over each completion Kv of K. In particular, the archimedean comple
tions of K lead back to classical dynamics over lR or In this chapter we describe
some of the fundamental concepts and theorems in complex dynamics. These the
orems provide both tools and templates for our subsequent study of dynamics over
complete nonarchimedean fields in Chapters 2 and 5 and over number fields in Chap
ters 3 and 4. This chapter provides sufficient background and motivation for reading
the remainder of this book, but the reader wishing to learn more about the beautiful
subject of complex dynamics might look at one or more of the following texts:
A. Beardon [43], Iteration ofRational Functions.
L. Carleson and T. Gamelin [95], Complex Dynamics.
R. Devaney [132], An Introduction to Chaotic Dynamical Systems.
J. Milnor [302], Dynamics in One Complex Variable.
C (
C.
ã
ã
ã
ã
1.1
Rational Maps and the Proj ective Line
A rationalfunction
Â(z) E C(z) is a quotient of polynomials
ao + a1 z + . . . + adzdd
=
¢(z) = F(z)
G(z) b0 + b1 z + + bdz
·
9
· ·
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1. An Introduction to Classical Dynamics
10
with no common factors. 1 The degree of¢ is
deg ¢ = max{deg F, deg G}.
ad bd
So if we write ¢ as above with at least one of and nonzero, then ¢ has degree d.
We generally consider maps with deg ¢ :::=: 2.
A rational function of degree d induces a rational map of the complex projective
linelP 1 (C),
( 1 . 1)
The map ¢ is a ramified d-to-1 covering. The complex projective line is also known
as the Riemann sphere, since topologically, and indeed real analytically, lP1 (C) is
isomorphic to S2 . Two very important properties of rational maps ( l . l ) are that they
are continuous and open. 2
There are many equivalent (and useful) ways to construct the projective line,
including
2
�
JP (C)
and
'
rv
where [ X,Y] "'[ X',Y'] ifthere is a u E
JP1 is given by a pair
ofhomogeneous polynomials ¢(X,Y) = [F* (X,Y), G* (X,Y)] of degree d. These
are related to the description
= F(z)/G(z) by the relations
-->
F* (X,Y) =
¢(z)
ydF(X/Y)
and
G* (X,Y) =Y d G(X/Y).
A linear fractional transformation (or Mobius transformation) is a map of the
form
z r---t
az + b
cz + d
with ad -
--
be # 0.
It defines an automorphism of JP 1 , and composition of transformations corresponds
to multiplication of the corresponding matrices ( � �). Two matrices give the same
transformation if and only if they are scalar multiples of one another, and it is not hard
to prove that these are the only automorphisms oflP1 (C); see [ 1 98, Exercise 1.6.6 and
Remark 11.7. 1 .1]. So we have
We observe that given any two triples ( a, a', a" ) and ((3, (3', (3" ) of distinct points
in JP 1 , there exists a unique automorphism E PGL2 (C) satisfying
f( a )
= (3,
f( a'
f
) = (3',
"
f( a ) = (3" .
1 It is permissible to have G(z)
0, in which case F(z) = ao is a nonzero constant and </>(z) maps
every point to oo. Similarly we may have F(z)
0 and G(z) = bo f- 0. Note that by convention we
set deg 0 -oo, so if ¢>(z) is constant, then deg ¢> 0.
2 Let ¢> : X --+ Y be a map between topological spaces. We recall that </> is continuous if U C Y open
implies </>- 1 (U) C X open, and ¢> is open ifV C X open implies </>(V) C Y open.
=
=
=
=
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11
1.1. Rational Maps and the Projective Line
PGL2 (C)
PGL2 (C),
This is intuitively reasonable because
is a three-dimensional space that
acts more-or-less simply transitively on JP>1 x JP>1 x JP>1 .
If ¢ JP> 1 JP> 1 is a rational map and f E
the linear conjugate of¢
by f is the map
:
-+
¢f
=
J - 1 0 ¢0 f.
Linear conjugation corresponds to a change of variables on JP>1 as illustrated by the
commutative diagram
f
1
]p> 1
f
�
1
]p> 1
Two rational maps ¢ and '1/J are linearly cmy'ugate if '1/J = ¢f for some f E
This is clearly an equivalence relation. We also observe that
PG12 (C).
which shows that linear conjugation is a good operation to use when studying itera
tion.
A convenient metric on the projective line JP>1 is the chordal metric, given in
homogeneous coordinates by the formula
(C)
p ([ X1 , y1 l '[ X2 ,
IX1Y - X Y11
2l)- J[X [2 + I Y 2[2J[X2 [2 + [Y [2
1
1
2
2
y;
If neither point is the point at infinity (i.e., neither point is equal to [ 1 , 0]), then using
the substitution zi = XdY;, to dehomogenize gives
p
l z2 l
p(z1, z2) = - lZ= 2 ;;=z+= 1::::olc-Jr,= lZ=2c;;;= l 2 =+=:=1
c;
l
1
Vr,=
1. Identifying IP' 1 9! U { oo} with the sphere S2 by drawing
(C) C
Notice that 0 � �
lines from the north pole of the sphere (see Figure 1 . 1), the chordal metric
is
related to the Euclidean distance between the corresponding points on the sphere by
the formula
p(z, w)
p(z,w) � lz*- w*[.
=
(See Exercise 1 . 1 .) In particular, the triangle inequality in JR3 implies the triangle
inequality for the chordal metric
(1 .2)
Rational maps ¢ IP' 1
IP'1 also have the Lipschitz property relative to
the chordal metric. This means that there is a constant ¢) such that
(C) (C)
p(¢(a), ¢(f3)) � C(¢)p(a,(3)
:
(See Exercise 1 .3.)
-+
C(
( 1 .3)
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12
1. An Introduction to Classical Dynamics
Figure 1 . 1 : Identifying
Critical Points and the Riemann-Hurwitz
Formula
¢(z) = F(z)/G(z), the formal derivative
¢'(z) = G(z)F'(z)G(z)- F(z)G'(z)
2
is defined at any point a satisfying a
and G(a) 0. (In order to compute
the derivative at the excluded points, make a linear change of variables. See Exer
cise 1 .5.)
Looking locally around z = a, Taylor's theorem says that
¢(z) = ¢(a) + ¢'(a)(z- a) + O((z - a)2 ).
lP' 1 (
talk about ramification points; dynamicists talk about critical points.) Note that ¢ is
Given a rational function
#
:
oo
#
---+
locally one-to-one around a noncritical point. As we will see, the orbits of the critical
points play an important role in determining the behavior of the dynamical system.
Assuming as above that -I oo and
-I oo, the ramification index of ¢ at
is
a
¢(a)
a
en(¢) = ordn(¢(z) - ¢(a)).
In particular, ¢ is ramified at a if and only if en ( ¢) ;::: 2. Locally, there is a constant
c # 0 such that
¢(z) = ¢(a) + c(z - a)ea(<l>) + 0 ((z - a)ea(<l>)+l ) '
and ¢ is locally en ( ¢)-to-1 in a neighborhood of a. It is not hard to see that eq,( a) �
deg(¢). If eq,(a) = deg(¢), then we say that ¢ is totally ramified at
a.
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1.2. Critical Points and the Riemann-Hurwitz Formula
13
The next result gives a global relation satisfied by the locally defined ramification
indices. We sketch two distinct proofs of this important theorem.
Theorem 1.1. (Riemann-Hurwitz Formula for 1P' 1 )
2d - 2
= aEII'L1 (ea(¢) - 1).
Proof. (Algebraic Proof) After a change of variables using a linear fractional trans
formation, we may assume that oo is neither a ramification point nor the image of a
ramification point, and also that ¢( oo ) 0. This means that ¢( has the form
=
F(z) azd-d l
¢( z) = G(z) bz
+
z)
·
· ·
= --::--;---
+..·
ab =/:- 0. Then
2d - 2 + . . .
G'(z)
G(z)F'(z) - F(z)
-abz
¢' ( z )
G(z)2
G(z)2
For any a =1- such that ¢(a) =f. i.e., such that G(a) =f. 0, we have
¢(z) = ¢(a) + (z - a)ea (<l>l�(z)
for a function �(z) E C(z) satisfying �(a) =f. 0, Differentiating yields
¢'(z) ea (</>) (z - a) e" (¢)-l�(z) + (z - a) ea (<l> )�'(z),
from which we deduce
orda ¢'(z) = ea (¢) - 1.
(Notice we are using strongly the fact that C has characteristic 0.) Summing over a
gives
L (ea(¢) - 1) L orda ¢' (z) = deg( -abz2d-2 + ) = 2d- 2.
a,¢(a)-f.=
a,¢(a)-f.=
(Topological Proof) It is a well-known topological fact that if a sphere is triangulated
with V vertices, E edges, and F faces, then V - E F 2. We take a (sufficiently
small) triangulation of the sphere with the property that the image of every ramifica
with
=
=
oo
oo,
oo.
=
=
·
+
·
·
=
tion point of ¢ is a vertex. Let
{vi :
1�i�
V}, {ei : 1 � i � E}, {fi : 1 � i � F},
be the vertices, edges, and faces of the triangulation. The map ¢ is a local isomor
phism except around the ramification points, so
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1. An Introduction to Classical Dynamics
14
E', F'
also gives a triangulation of the sphere. To ease notation, we let V', and denote
the numbers of vertices, edges, and faces in this inverse image triangulation.
The map ¢ is exactly d-to-1 except over the ramification points, so the set
consists of exactly dE edges, and similarly the set
consists of exactly dF faces. In other words, = d and =
The vertices are a little more complicated. If v is a vertex, then ¢-1 ( v) con
sists of d points counted with appropriate multiplicities, where the multiplicity
of a E ¢- 1 ( v) is precisely (¢). In other words,
E' E F' dF.
ea
= a 2::1 v ea(¢),
E¢- ( )
d - #¢-1 (v) = 2:: (ea (¢) - 1).
aEc/>-1 (v)
or equivalently,
d
Summing over the vertices Vi yields
dV-
v
2:: #¢- 1 (vi)
i=l
v
= 2::
2::
(ea(¢) - 1).
Note that the lefthand side is dV- V'. Further, since we chose the Vi 's to include the
image of every critical point, we know that (¢) = 1 for every point a not in any
of the ¢- 1 (vi ) 's, so we can extend the sum on the righthand side to include every
a E !P' 1
This proves that
ea
(C).
dV = V' +
2:::
aEII'1 (rC)
(ea (¢) - 1).
Next we take an alternating sum to obtain
d(V-
E + F) = V'- E' + F' + aEII'2::1 rC (ea(¢) - 1).
()
Finally, we use the fact that
V
-
E + F = V' - E' + F' = 2
to obtain the Riemann-Hurwitz formula.
D
Corollary 1.2. A rational map ofdegree d has exactly 2d - 2 critical points, counted
with appropriate multiplicities.
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15
1.2. Critical Points and the Riemann-Hurwitz Formula
The following weaker version of the Riemann-Hurwitz formula is often conve
nient for applications.
¢ : JP>1 (
JP>1 (C).
Corollary 1.3. Let
q
(a) Let a E
Then
-->
JP>1 (q be a rational map ofdegree d ?: 1.
L ef3(¢) = d.
{3E¢-1(a)
(b) [Weak Riemann-Hurwitz Formula]
2d - 2 = L (d - #¢- 1 (a)).
aEIP'1(C)
a
Proof Making a change of coordinates reduces us to the case that =1- oo and
oo �
Writing
and
{,61, ... , ,6r }, this means
¢- 1 (a). ¢(z) = F(z)/G(z) ¢- 1 (a) =
that there is a factorization
ei = ef3; (¢), so
L ef3(¢) = Lr ei = deg(¢) = d.
i=l
Notice that the exponents are exactly the ramification indices
{3E¢-1(a)
This proves (a). We then use the Riemann-Hurwitz formula (Theorem 1 . 1) to com
pute
= L (d - #</>- 1 (a)) .
aEIP'1(C)
D
Remark 1 .4. The Riemann-Hurwitz formula is an example of a local-global for
2d-2
mula. The quantity
is a global quantity, given in terms of how many times the
map covers the Riemann sphere lP'1 (q and reflecting the topology of the sphere.
The general version of the Riemann-Hurwitz formula, which we now state, includes
global information about the genera of the curves under consideration.
</>
Theorem 1.5. (Riemann-Hurwitz Formula) Let C1 and C2 be algebraic curves (Rie
mann surfaces) ofgenus and respectively, and let ¢ : cl --) c2 be a.finite map
ofdegree
1. Then
d ?:
91 92,
291 - 2 = d(292- 2) + L (ep(Â) - 1).
PEC1
Proof See [198, IV Đ2].
D
