Author | Ioannis Farmakis | |

Isbn | 9789814458917 | |

File size | 1MB | |

Year | 2013 | |

Pages | 240 | |

Language | English | |

File format | ||

Category | mathematics |

FIXED POINT THEOREMS AND
THEIR APPLICATIONS
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2/7/13 12:02 PM
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FIXED POINT THEOREMS AND
THEIR APPLICATIONS
Ioannis Farmakis
Martin Moskowitz
City University of New York, USA
World Scientific
NEW JERSEY
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LONDON
8748_9789814458917_tp.indd 2
•
SINGAPORE
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BEIJING
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SHANGHAI
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HONG KONG
•
TA I P E I
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CHENNAI
2/7/13 12:02 PM
Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Cover: Image courtesy of Jeff Schmaltz, MODIS Land Rapid Response Team at NASA GSFC.
FIXED POINT THEOREMS AND THEIR APPLICATIONS
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-981-4458-91-7
Printed in Singapore
Contents
Preface and Acknowledgments
Introduction
ix
1
1 Early Fixed Point Theorems
1.1 The Picard-Banach Theorem . . . . . . . . . . . . . . .
1.2 Vector Fields on Spheres . . . . . . . . . . . . . . . . . .
1.3 Proof of the Brouwer Theorem and Corollaries . . . . .
1.3.1 A Counter Example . . . . . . . . . . . . . . . .
1.3.2 Applications of the Brouwer Theorem . . . . . .
1.3.3 The Perron-Frobenius Theorem . . . . . . . . . .
1.3.4 Google; A Billion Dollar Fixed Point Theorem . .
1.4 Fixed Point Theorems for Groups of Affine Maps of Rn
1.4.1 Affine Maps and Actions . . . . . . . . . . . . . .
1.4.2 Affine Actions of Non Compact Groups . . . . . .
3
3
5
9
12
15
16
21
24
25
30
2 Fixed Point Theorems in Analysis
2.1 The Scha¨
uder-Tychonoff Theorem . . . . . . . . . . . .
2.1.1 Proof of the Scha¨
uder-Tychonoff Theorem . . . .
2.2 Applications of the Scha¨
uder-Tychonoff Theorem . . . .
2.3 The Theorems of Hahn, Kakutani and Markov-Kakutani
2.4 Amenable Groups . . . . . . . . . . . . . . . . . . . . .
2.4.1 Amenable Groups . . . . . . . . . . . . . . . . . .
2.4.2 Structure of Connected Amenable Lie Groups . .
35
36
38
43
46
51
52
54
v
vi
3 The Lefschetz Fixed Point Theorem
3.1 The Lefschetz Theorem for Compact Polyhedra .
3.1.1 Projective Spaces . . . . . . . . . . . . . .
3.2 The Lefschetz Theorem for a Compact Manifold
3.2.1 Preliminaries from Differential Topology .
3.2.2 Transversality . . . . . . . . . . . . . . . .
3.3 Proof of the Lefschetz Theorem . . . . . . . . . .
3.4 Some Applications . . . . . . . . . . . . . . . . .
3.4.1 Maximal Tori in Compact Lie Groups . .
3.4.2 The Poincar´e-Hopf’s Index Theorem . . .
3.5 The Atiyah-Bott Fixed Point Theorem . . . . . .
3.5.1 The Case of the de Rham Complex . . . .
Contents
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57
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4 Fixed Point Theorems in Geometry
4.1 Some Generalities on Riemannian Manifolds . . . . . . .
4.2 Hadamard Manifolds and Cartan’s Theorem . . . . . . .
4.3 Fixed Point Theorems for Compact Manifolds . . . . . .
109
110
124
135
5 Fixed Points of Volume Preserving Maps
5.1 The Poincar´e Recurrence Theorem . . . . . . . . . . . .
5.2 Symplectic Geometry and its Fixed Point Theorems . .
5.2.1 Introduction to Symplectic Geometry . . . . . . .
5.2.2 Fixed Points of Symplectomorphisms . . . . . . .
5.2.3 Arnold’s Conjecture . . . . . . . . . . . . . . . .
5.3 Poincar´e’s Last Geometric Theorem . . . . . . . . . . .
5.4 Automorphisms of Lie Algebras . . . . . . . . . . . . . .
5.5 Hyperbolic Automorphisms of a Manifold . . . . . . . .
5.5.1 The Case of a Torus . . . . . . . . . . . . . . . .
5.5.2 Anosov Diffeomorphisms . . . . . . . . . . . . . .
5.5.3 Nilmanifold Examples of Anosov Diffeomorphisms
5.6 The Lefschetz Zeta Function . . . . . . . . . . . . . . .
143
143
146
146
153
154
155
163
167
169
173
177
179
6 Borel’s Fixed Point Theorem in Algebraic Groups
187
6.1 Complete Varieties and Borel’s Theorem . . . . . . . . . 187
6.2 The Projective and Grassmann Spaces . . . . . . . . . . 190
6.3 Projective Varieties . . . . . . . . . . . . . . . . . . . . . 193
vii
Contents
6.4
6.5
Consequences of Borel’s Fixed Point Theorem . . . . . . 197
Two Conjugacy Theorems for Real Linear Lie Groups . 200
7 Miscellaneous Fixed Point Theorems
7.1 Applications to Number Theory . . . . . . . .
7.1.1 The Little Fermat Theorem . . . . . .
7.1.2 Fermat’s Two Squares Theorem . . . .
7.2 Fixed Points in Group Theory . . . . . . . . .
7.3 A Fixed Point Theorem in Complex Analysis
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203
203
203
205
207
209
8 A Fixed Point Theorem in Set Theory
211
Afterword
217
Bibliography
219
Index
229
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Preface and
Acknowledgments
Our intention here is to show the importance, usefulness and pervasiveness of fixed point theorems in mathematics generally and to try to do
so by as elementary and self contained means as possible. The book
consists of eight chapters.
Chapter 1, Early fixed point theorems, deals mostly with matters
connected to the Brouwer fixed point theorem and vector fields on
spheres, the Contraction Mapping Principle and affine mappings of finite dimensional spaces. As we shall see, the Brouwer theorem can be
generalized in basically two ways, one leading to the Lefschetz fixed
point theorem in topology (whose proof is independent of Brouwer) and
the other to the Scha¨
uder-Tychonoff fixed point theorem of analysis
(which depends on the Brouwer theorem). We shall draw a number
of consequences of Brouwer’s theorem: among them Perron’s theorem
concerning eigenvalues of a “positive” operator, which itself has interesting applications to the modern world of computing such as in Google
addresses and ranking in professional tennis.
Chapter 2, Fixed point theorems in Analysis, deals with the theorems
of Scha¨
uder-Tychonoff, as well as the those of F. Hahn, Kakutani and
Kakutani-Markov involving groups of affine mappings, the latter two
leading naturally to a discussion of Amenable groups.
Chapter 3 concerns Fixed point theorems in Topology, particularly
the Lefschetz fixed point theorem, the H. Hopf index theorem and some
of their consequences. As a further application we prove the conjugacy
ix
x
Preface and Acknowledgments
theorem for maximal tori in a compact connected Lie group. In addition,
we explain the Atiyah-Bott fixed point theorem and its relationship to
the classical Lefschetz theorem.
Chapter 4, Fixed point theorems in Geometry, is devoted first to
the fixed point theorem of E. Cartan on compact groups of isometries
of Hadamard manifolds and then to fixed point theorems for compact
manifolds, when the curvature is negative, due to Preissmann, and positive, due to Weinstein.
Chapter 5 concerns Fixed points of maps preserving a volume form,
which are important in dynamical systems. We begin with the Poincar´e
recurrence theorem and deal with some of its philosophical implications. Then we turn to symplectic geometry and fixed point theorems
of symplectomorphisms. Here we discuss Arnold’s conjecture, prove
Poincar´e’s last geometric theorem and derive a classical result concerning billiards. We then turn to hyperbolic automorphisms of a compact
manifold, particularly of a torus, and then to Anosov diffeomorphisms
and the analogous Lie algebra automorphisms and explain their significance in dynamics. We conclude this chapter with the Lefschetz zeta
function and its many applications.
Chapter 6 deals with the Fixed point theorem of A. Borel for a solvable algebraic group acting on a complex projective variety, the most
important such varieties being the Grassmann and flag varieties. We
then present some consequences of these ideas. A final section concerns
a fixed point theorem giving rise to a conjugacy theorem for unipotent
subgroups of real reductive linear Lie groups.
We conclude with two brief chapters. Chapter 7 deals with some
connections of fixed points to number theory, group theory and complex
analysis, while in Chapter 8, A fixed point theorem in Set Theory, we
prove Tarski’s fixed point theorem and, as a consequence, the Schr¨oderCantor-Bernstein theorem.
As the reader can see, fixed point theorems are to be found throughout mathematics. These chapters can, for the most part, be read independently. Thus the reader has many options to follow his or her
particular interests.
Preface and Acknowledgments
xi
We thank Oleg Farmakis for his help in creating the diagrams and
Hossein Abbaspour for reading an earlier draft of the manuscript and
making a number of useful suggestions for improvements. Of course,
any mistakes are the responsibility of the authors. Finally, we thank
Konstantina and Anita for their patience.
New York, January 2013
Ioannis Farmakis, Martin Moskowitz
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Introduction
As exactly a century has passed since the Brouwer fixed point theorem
[22] was proved and because in some sense this result is the progenerator
of some of the others, it seems appropriate to consider fixed point theorems afresh and in general. As we shall see, these are both quite diverse
and pervasive in mathematics. Fixed point theorems are to be found in
algebra, analysis, geometry, topology, dynamics, number theory, group
theory and even set theory. Before proceeding it would be well to make
precise what we mean by a fixed point theorem.
Definition 0.1.1. Let X be a set and f : X → X be a map from X to
itself. A point x ∈ X is called a fixed point of f if f (x) = x.
A fixed point theorem is a statement that specifies conditions on X and
f guaranteeing that f has a fixed point in X. More generally, we shall
sometimes want to consider a family F of self maps of X. In this context
F is usually a group (or sometimes even a semigroup, see [82]). In this
case a fixed point theorem is a statement that specifies conditions on X
and F guaranteeing that there is a simultaneous fixed point, x ∈ X, for
each f ∈ F.
When F is a group G this will usually arise from a group action
φ : G × X → X of G on X. We will write φ(g, x) := gx. We shall
assume that the 1 of G acts as the identity map of X and for all g, h ∈ G
and x ∈ X that (gh)x = g(hx). We shall call such an X, a G-space.
Particularly, when X is a topological space and G is a topological group,
we shall assume φ is jointly continuous.
1
2
Introduction
Now, as above, when we have an action of a group G on Y, by taking
for “X” the power set, P(Y ), this induces an action of G on P(Y ) and
the fixed points of this new action are then precisely the G-invariant
subsets of Y . In this way, in many contexts, fixed point theorems give
rise to G-invariant sets.
We conclude this introduction with the remark (and example) that
the existence of Haar measure µ on a compact topological group G can
be regarded as a fixed point theorem. This is because in the associated
action of G by left translation on the space M + (G) of all finite positive
measures on G, left invariance simply means µ is G-fixed. Moreover,
since µ is finite and can be normalized so that µ(G) = 1, we can even
regard it as a G-fixed point under the action of G on the convex (and
compact in the weak∗ topology) set of positive normalized measures on
G (see Corollary 2.3.5).
Chapter 1
Early Fixed Point
Theorems
1.1
The Picard-Banach Theorem
One of the earliest and best known fixed point theorems is that of
Picard-Banach 1.1.1. Either explicitly or implicitly this theorem is the
usual way one proves local existence and uniqueness theorems for systems of ordinary differential equations (see for example [79], sections 7.3
and 7.5). This theorem also can be used to prove the inverse function
theorem (see [79], pp. 179-181).
Theorem 1.1.1. Let (X, d) be a complete metric space and f : X → X
be a contraction mapping, that is, one in which there is a 1 > b > 0
so that for all x, y ∈ X, d(f (x), f (y)) ≤ bd(x, y). Then f has a unique
fixed point.
Proof. Choose a point x1 ∈ X (in an arbitrary manner) and construct
the sequence xn ∈ X by xn+1 = f n (x1 ), n ≥ 2. Then xn is a Cauchy
sequence. For n ≥ m,
d(xn , xm ) = d(f n (x1 ), f m (x1 )) ≤ bm d(f n−m (x1 ), x1 ).
But
d(f n−m (x1 ), x1 ) ≤ d(f n−m (x1 ), f n−m−1 (x1 )) + . . . + d(f (x1 ), x1 ).
3
4
Chapter 1 Early Fixed Point Theorems
The
term is ≤ (bn−m−1 + . . . + b + 1)d(f (x1 ), x1 ) which is itself
Platter
∞
≤ n=0 bn d(f (x1 ), x1 ). Since 0 < b < 1 this geometric series converges
1
to 1−b
d(f (x1 ), x1 ). Since bm → 0 we see for n and m sufficiently large,
given ǫ > 0,
1
d(xn , xm ) ≤ bm
d(f (x1 ), x1 ) < ǫ.
1−b
Hence xn is Cauchy. Because X is complete xn → x for some x ∈ X.
As f is a contraction map it is (uniformly) continuous. Hence f (xn ) →
f (x). But as a subsequence f (xn ) → x so the uniqueness of limits tells
us x = f (x).
Now suppose there was another fixed point y ∈ X. Then
d(f (x), f (y)) = d(x, y) ≤ bd(x, y) so that if d(x, y) 6= 0 we conclude
b ≥ 1, a contradiction. Therefore d(x, y) = 0 and x = y.
We remark that the reader may wish to consult Bessaga ([11]), or
Jachymski, ([60]) where the following converse to the Picard-Banach
theorem has been proved.
Theorem 1.1.2. Let f : X → X be a self map of a set X and 0 < b < 1.
If f n has at most 1 fixed point for every integer n, then there exists a
metric d on X for which d(f (x), f (y)) ≤ bd(x, y), for all x and y ∈ X.
If, in addition, some f n has a fixed point, then d can be chosen to be
complete.
As a corollary to the Picard-Banach theorem we have the following
precursor to the Brouwer theorem.
Corollary 1.1.3. Let X be the closed unit ball in Rn and f : X → X
be a nonexpanding map; that is one that satisfies d(f (x), f (y)) ≤ d(x, y)
for all x, y ∈ X. Then f has a fixed point.
Proof. For positive integers, n, define fn (x) = (1− n1 )f (x). Then each fn
is a contraction mapping of X. Since X is compact, it is complete. By
the contraction mapping principle each fn has a fixed point, xn ∈ X and
since X is compact xn has a subsequence converging to say x. Taking
limits as n → ∞ shows x is fixed by f .
We now state the Brouwer fixed point theorem.
1.2
Vector Fields on Spheres
5
Theorem 1.1.4. Any continuous map f of the closed unit ball B n in
Rn to itself has a fixed point.
In other words, if one stirs a mug of coffee then at any given time
there is at least one particle of coffee that is in exactly the position it
started in.
Of course, since the Brouwer theorem is stated in purely topological
terms, it is actually true for any topological space X homeomorphic to
B n ; so in particular, for any compact convex set in Rn . (As an exercise
the reader might want to prove this.) Notice that as compared to the
Picard-Banach theorem, here we have no uniqueness. Also, the space is
rather specific, for example it is compact (and convex), but the map is
rather general. In Picard-Banach it is the opposite. The space is merely
complete, but the map is required to have very strong properties and
so we get uniqueness. In the Brouwer theorem there is no uniqueness.
For example the identity map has all of B n as fixed points.
To illustrate the Brouwer theorem in dimension one, consider a
square (with boundary) S in the plane and the diagonal going from
lower left to upper right. Choose two points, one on the left edge of S
and one on the right. Then the result says that any continuous curve
joining these points and lying wholly in the square must intersect the
diagonal.
One way to prove the Brouwer fixed point theorem is by studying
vector fields on Spheres (Theorem 1.2.1). In Chapter 2 we shall see that
this result is also a consequence of the Lefschetz fixed point theorem
and that it also implies Theorem 1.2.1.
1.2
Vector Fields on Spheres
Let X be a compact submanifold of Euclidean space Rn+1 . A continuous
(resp. smooth) tangential vector field v on X is a continuous (resp.
smooth) Rn+1 -valued function on X with the property that for each
x ∈ X, v(x) is tangent to X at x, that is it lies in the tangent hyperplane,
Tx (X). We shall call v nonsingular if v(x) is never 0 at any point of
X. For example, if X = S 2 , the 2-sphere, considered as the surface of
6
Chapter 1 Early Fixed Point Theorems
the earth, then the tangential component of the wind gives a tangential
vector field (which one assumes is continuous, but which may turn out
to be singular). Given a nonsingular smooth, or continuous tangential
vector field v on X, we can produce a unitary continuous tangential
vector field w on X, that is, one whose vectors are all of unit length,
v(x)
simply by normalizing v at each point: w(x) = kv(x)k
, x ∈ X. One
checks easily that w is smooth (resp. continuous). An example of a
nonsingular (unit) tangent vector field on S 1 is given by choosing any
point, x0 , and placing a unit vector v0 at x0 tangent to the circle there.
Then since any other point x ∈ S 1 can be obtained from x0 by a unique
counter clockwise rotation we just rotate v0 by the same amount and
get a unit tangent vector to x. Here we are particularly interested in
the unit sphere, X = S n ⊆ Rn+1 . In this case each tangent vector v(x)
is perpendicular to x, x ∈ S n . That is hx, v(x)i = 0 on S n . One verifies
easily that this condition is both necessary and sufficient for tangency.
The first question we address is: are there nonsingular smooth (or
continuous) vector fields on S n ? When n = 2k − 1 is odd there always
are. For example, we will show that for x = (x1 , . . . , x2k ) ∈ S 2k−1 ,
v(x1 , . . . , x2k ) = (x2 , −x1 , x4 , −x3 , . . . , x2k , −x2k−1 )
is such a vector field. For v is evidently a smooth function of x. To
see that it is unitary, notice v(x1 , . . . , x2k ) = xe − xo , where xe is the
vector in R2k with even subscripts x2i in the odd coordinates and zeros
in the even ones, while xo is the vector in R2k with zeros in the odd
coordinates and x2i−1 in the even ones. Hence
k v(x) k2 = hxe − xo , xe − xo i = k xe k2 + k xo k2 + 2hxe , xo i.
However, hxe , xo i = 0 and k xe k2 + k xo k2 = k x k2 = 1. Therefore,
k v(x) k2 ≡ 1. Moreover, one sees immediately using the fact that we
are in an even dimensional Euclidean space that hv(x), xi = 0. Thus v
is a smooth unitary tangential vector field on X.
Notice that one cannot even write such a thing when n is even.
Indeed as we shall see, when n is even there are no continuous nonsingular vector fields on X. So for example, at any moment there must be
1.2
Vector Fields on Spheres
7
a point on the earth’s surface where the wind does not blow tangentially.
Viewed from the aspect of vector fields it is this fact which lies behind
the Brouwer theorem.
In this section we prove the following result, Theorem 1.2.1 (sometimes called the hairy ball theorem). Then, we shall extend Theorem 1.2.1 from the smooth to the continuous case from which we will
get the Brouwer fixed point theorem 1.1.4. Finally, from Theorem 1.1.4
as a corollary we will get the Invariance of Domain theorem 1.3.1 and
other applications.
Theorem 1.2.1. S n possesses a nonsingular smooth vector field if and
only if n is odd.
Since we have shown just above that odd dimensional spheres always
have a nonsingular smooth vector field, it remains to prove that if n is
even there can be no such vector field.
Let n = 2k, i.e. n is even and suppose there were a smooth unitary
tangential vector field v on S n . For t ∈ R let Ft (x) = x + tv(x), for all
x ∈ S n . We calculate k x + tv(x) k. Just as above
hx + tv(x), x + tv(x)i = hx, xi + 2thx, v(x)i + t2 hv(x), v(x)i.
Since both k x k = k v(x) k = 1 and hx, v(x)i = 0, we see that
p
k x + tv(x) k = 1 + t2 .
To complete the proof of our theorem we shall need the following two
lemmas.
Lemma 1.2.2. If |t| is sufficiently small, Ft is a 1 : 1 mapping taking
X onto a region whose volume can be expressed as a polynomial function
of t.
Let vol∗ denote the n-dimensional volume (as opposed to the n + 1
dimensional volume in Rn+1 ).
Lemma 1.2.3. If |t| is sufficiently small, the map Ft takes S n onto the
√
n+1
sphere of radius 1 + t2 . Hence vol∗ (Ft (S n )) = (1 + t2 ) 2 vol∗ (S n ).

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