Fixed Point Theorems and Their Applications by Ioannis Farmakis


7858edc85045bd6-261x361.jpeg Author Ioannis Farmakis
Isbn 9789814458917
File size 1MB
Year 2013
Pages 240
Language English
File format PDF
Category mathematics


 

FIXED POINT THEOREMS AND THEIR APPLICATIONS 8748_9789814458917_tp.indd 1 2/7/13 12:02 PM This page intentionally left blank FIXED POINT THEOREMS AND THEIR APPLICATIONS Ioannis Farmakis Martin Moskowitz City University of New York, USA World Scientific NEW JERSEY • LONDON 8748_9789814458917_tp.indd 2 • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI 2/7/13 12:02 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 . 58 . 62 . 64 . 64 . 67 . 76 . 81 . 83 . 87 . 94 . 104 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 203 203 205 207 209 8 A Fixed Point Theorem in Set Theory 211 Afterword 217 Bibliography 219 Index 229 This page intentionally left blank 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 This page intentionally left blank 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 ).

Author Ioannis Farmakis Isbn 9789814458917 File size 1MB Year 2013 Pages 240 Language English File format PDF Category Mathematics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This is the only book that deals comprehensively with fixed point theorems overall of mathematics. Their importance is due, as the book demonstrates, to their wide applicability. Beyond the first chapter, each of the other seven can be read independently of the others so the reader has much flexibility to follow his/her own interests. The book is written for graduate students and professional mathematicians and could be of interest to physicists, economists and engineers. Readership: Graduate students and professionals in analysis, approximation theory, algebra and geometry.     Download (1MB) Renormalization and Effective Field Theory A Guide To Nip Theories K-theory For Operator Algebras Introduction To Ramsey Spaces Forcing Idealized (cambridge Tracts In Mathematics) Load more posts

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