Continuous Dynamics On Metric Spaces by Craig Calcaterra

8/14/2019 Continuous Dynamics On Metric Spaces by Craig Calcaterra

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Continuous Dynamicson Metric Spaces

Craig Calcaterra

29 November 2008Version 1.0


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Contents

Preface v

Introduction vii0.1 Context and objective . . . . . . . . . . . . . . . . . . . . . . . . vii0.2 Example: ows on L2 (R) . . . . . . . . . . . . . . . . . . . . . . xi0.3 Example: ows on manifolds . . . . . . . . . . . . . . . . . . . . xiv0.4 Example: ows on a space with no linear structure . . . . . . . . xxi0.5 Chapter outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii0.6 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv0.7 Abridged version of the book . . . . . . . . . . . . . . . . . . . . xxv0.8 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi

I Theory 1

1 Flows 31.1 Generating ows with arc elds . . . . . . . . . . . . . . . . . . . 3

1.1.1 The fundamental theorem . . . . . . . . . . . . . . . . . . 31.1.2 Local ows . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.3 Global ows . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Forward ows and xed points . . . . . . . . . . . . . . . . . . . 191.3 Invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4 Commutativity of ows . . . . . . . . . . . . . . . . . . . . . . . 22

2 Lie algebra on metric spaces 252.1 Metric space arithmetic . . . . . . . . . . . . . . . . . . . . . . . 252.2 Metric space Lie bracket . . . . . . . . . . . . . . . . . . . . . . . 312.3 Covariance and contravariance . . . . . . . . . . . . . . . . . . . 34

3 Foliations 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Local integrability . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Commutativity of ows . . . . . . . . . . . . . . . . . . . . . . . 583.4 The Global Frobenius Theorem . . . . . . . . . . . . . . . . . . . 60

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iv CONTENTS

3.5 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

II Examples 71

4 Brackets on function spaces 73

5 Approximation with non-orthogonal families 835.1 Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 First approximation formula . . . . . . . . . . . . . . . . . 835.1.2 Signal synthesis . . . . . . . . . . . . . . . . . . . . . . . . 845.1.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . 855.1.4 Coefficient formulas . . . . . . . . . . . . . . . . . . . . . 885.1.5 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2 Low-frequency trigonometric series . . . . . . . . . . . . . . . . . 90

5.2.1 Density in L2 . . . . . . . . . . . . . . . . . . . . . . . . . 905.2.2 Coefficient formulas . . . . . . . . . . . . . . . . . . . . . 925.2.3 Damping gives a stable family . . . . . . . . . . . . . . . . 97

6 Partial differential equations 1016.1 Metric space arithmetic . . . . . . . . . . . . . . . . . . . . . . . 1016.2 PDEs as arc elds . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Flows on H (Rn ) 1077.1 IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Continuous IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.3 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4 Cyclically attracted sets . . . . . . . . . . . . . . . . . . . . . . . 114

7.5 Control theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158 Counter-examples 119

Appendix A: Metric spaces 123.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

.2.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

.2.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 130.3 Geometric objects . . . . . . . . . . . . . . . . . . . . . . . . . . 131

.3.1 Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

.3.2 Metric coordinates . . . . . . . . . . . . . . . . . . . . . . 132

.3.3 Conversion formulas . . . . . . . . . . . . . . . . . . . . . 133

Appendix B: ODEs as vector elds 137

Appendix C: Numerical differentiation 141

List of notation 151


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Preface

This book explores the subject of metric geometry using continuous dynamics.Metric geometry is currently experiencing intense interest, due to Perelmanssolution of the Poincares Conjecture and the inuence of Gromovs ideas onstring theory in physics. Despite this advanced pedigree, metric geometry beginsat a basic level requiring no more than an undergraduate introduction to pointset topology and the denition of a distance metric. The novel perspective of this text is the focus of using ows on an abstract metric space to crack intogeometric objects such as foliations. The abstract environment allows us topinpoint the necessary ideas to make all our analytic constructionswe employthe bare minimum denitions for creating dynamics, geometric decompositions,and approximations on metric spaces. This book is written with students inmind, with the intention of using this minimum apparatus to make learning andunderstanding the ideas easier. Hopefully the treatment will be of interest toresearchers as well, being the rst unied presentation of this dynamic approachto metric geometry. Further, researchers can use this abstract environment totest the limits of their understanding of fundamental constructions such as ows,

Lie derivatives, foliations, holonomy and connections.

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vi PREFACE


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Introduction

In this chapter the case is made for the importance of studying ows on a metricspace. The concept of a metric space is the deepest point of contact betweengeometry and analysis; we gain new perspective on these subjects by generalizingseveral of their results to metric spaces. The generalized Fundamental Theoremof Ordinary Differential Equations and Frobenius Foliation Theorem are themajor theoretical results of this book. The rst theorem belongs to analysisand the second to geometry.

The greater generality also gives a richer palette for mathematical modeling,as demonstrated with novel dynamics on H (Rn ), the space of nonempty com-pact subsets of Rn . Innovative dynamics arise even on well-studied spaces. E.g.,geometric control theory on function spaces leads to our centerpiece example:low-frequency trigonometric series can approximate any L2 function on any in-terval, Theorem 94 and Example 95, which the reader can turn to immediately,before learning the details of metric space dynamics which conceived the idea.

0.1 Context and objectiveA metric space (M, d ) is a set M with a function d : M M R called themetric which is positive, denite, symmetric and satises the triangle inequal-ity:

(i) d(x, y ) 0 positivity(ii) d(x, y ) = 0 iff x = y deniteness ( or non-degeneracy)(iii) d(x, y ) = d(y, x ) symmetry(iv) d(x, y ) d(x, z ) + d(z, y) triangle inequality

for all x,y,z M . A metric space is locally complete if for each elementxM there exists an r > 0 such that the closed ball

B (x, r ) := {yM |d (x, y ) r}is complete. Every major result in this book is written at this generality, so ourconstant friend is the triangle inequalityexploited without acknowledgement.The most important metric spaces include n-dimensional Euclidean space Rn ,Riemannian manifolds and function spaces such as L2 (R). Appendix A gives

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viii INTRODUCTION

denitions for these and other examples and lists general properties of metricspaces.

The term continuous dynamics, as opposed to discrete dynamics, meansthe study of ows:

Denition 1 A ow is a continuous map F : M R M which, for all xM and s, t R, satises (i) F (x, 0) = x(ii ) F (F (x, s ) , t ) = F (x, s + t).

More efficient notations are

F t (x) := F (x, t ) =: F x (t)

with the space variable x or time parameter t in the subscript, depending on

which quantity is active in a calculation. Flows will typically be denoted withF , G, or H .For xed t, a ow gives a map F t : M M which is necessarily an auto-morphism, i.e., a homeomorphism of M to itself, since F t is the continuous

inverse of F tF t F t = F 0 = Id .

A ow may thus be viewed as a 1-parameter family of homeomorphisms. Fromanother point of view, the R parameter t often signies time, and the map F x :R M then describes the motion of a xed x through its position/congurationspace M F x (t) for all tR with initial condition x = F x (0).

Our chief interest is to use continuous dynamics to explore the geometry

of general metric spaces. Insights into geometric structure, in turn, give usdeeper understanding of possible dynamics. It is surprising how many importantgeometrical ideas require only a metric for their denition. Balls and spheres,of course, are utilized at the inception of metric spaces. A more extensive listof static geometric denitions (ellipses, cylinders, etc.) appears on page 133.Ekelands variational principle and the Mountain Pass Theorem have naturalexpressions on a metric space [43]. For many decades algebraic topologistshave been aware that a topology without further algebraic structure is sufficientto dene geometrically insightful indices, such as the fundamental homotopygroup or the homological Conley index [25]. More important for this book,geometric notions such as curves, surfaces, tangency, and transversality havenatural expressions on metric spaces. The generalization of the FundamentalTheorem of Ordinary Differential Equations to metric spaces ([52], [7], [18], [30])

and Frobenius Foliation Theorem (Chapter 3) are the major theoretical resultsexplicated in this text. Further, length, speed, angles, norm, curvature [14],the Lie derivative (Chapter 2), gradients ([41], [3]) and many others also havenatural and fruitful generalizations. The spirit that guides the development of metric geometry is the conviction that every major geometrical result has asubstantial expression on metric spaces.


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x INTRODUCTION

space. For instance a curve c : I Rn is differentiable with c (t0) Rn fort0

I if and only if c is tangent to the curve l (t) := c (t0)+ ( t

t0) c (t0) which

is a line in the direction of c (t0) since

limh0

d (c (t0 + h) , l (t0 + h))h

= limh0

c (t0 + h) c (t0)h c (t0) (2)

where the metric d is derived from the norm, d (x, y ) := x y . So the smooth-ness of a curve c is determined by its tangency with a special curve, an arc, l.Remember (Appendix B) nearly any ODE may be rewritten as a vector eld

problemx = V (x)

where V : Rn Rn is the vector eld and a solution is a curve x 0 : I Rnwith initial condition x 0 (0) = x0Rn satisfyingddt x0 (t) = V (x 0 (t)) .

The fundamental result of ODEs is: if V is Lipschitz continuous then there existsa collection of solutions which generates a unique local ow F (x, t ) := x (t).We generalize this result in Chapter 1 using the idea contained in (2) that acurve can represent a vector or derivative. In analogy with vectors on a linearspace, we study arcs on a metric space. Whereas the vector eld V species adirection V (x)R

n at each point xRn to which solutions must be tangent,

an arc eld X species a direction with an arc at each point x. So an arc eldis a map X : M [1, 1] M with X (x) : [1, 1] M being the arc at theposition xM .To make the generalization claimed in the previous paragraph more concrete,let us show how every vector eld V may be naturally represented as an arc

eld X . Dene X : Rn

[1, 1] Rn

by X (x, t ) := x + tV (x). If x 0 : I Rn

is a solution to the vector eld problem, then x0 is also tangent to X at eachvalue tI in the sense that

limh0

d ( (t + h) , X ( (t) , h))h

= 0 .

To check this notice

limh0

d ( (t + h) , X ( (t) , h))h

= limh0

d ( (t + h) , (t) + hV ( (t)))h

= limh0

(t + h) (t)h V ( (t))

=ddt

(t) V ( (t)) = 0 .

The motivation for generalizing the calculus is to analyze dynamics (i.e.,ows) on such archetypical examples of metric spaces as the innite-dimensionalspace L2 (R), manifolds, and the space of non-empty compact subsets of theplane H R2 .


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0.2. EXAMPLE: FLOWS ON L2 (R) xi

0.2 Example: ows on L 2 (R)The space of square integrable functions L2 (R) (see Appendix A.1 for a precisedenition) is a linear space and may seem an unlikely candidate to yield novelresults through our program of abstracting classical results to metric spaceswhile avoiding the use of any linear structure. However, for this most elemen-tary of all innite-dimensional spacesthis Hilbert spacethe linear structure isactually a hindrance to understanding some of its most basic ows.

Example 2 On M := L2 (R) , the (Hilbert ) space of square integrable functions of one real variable, the metric is derived from the L2 norm :

d (f, g ) := R (f g)2 d = f g 2 .What is the simplest example of a ow on M ? For many visual thinkers, trans-lating the graph leaps to mind :

F (f, t ) (x) := f (x + t) .

f (x + t) and f (x)

The two ow properties are automatically veried : (i) F (f, 0) = f and (ii )F (F (f, s ) , t ) = F (f, s + t) for any f L

2 (R). In fact {F (, t ) |tR} is clearly a family of isometries of M .This example seems so perfectly regular as to seem trivial. However a con-

founding blow to our intuition is that for most initial conditions f , the curvesF (f, ) are non-differentiable with respect to either Gateaux or Frechet differen-tiability (notions we wont use and wont dene). To get a feel for this situation,consider the initial condition f := [0,1] . Here S represents the characteristicfunction of a set S , i.e.,

S (x) := 1 for xS 0 otherwise.

For f := [0,1]

F (f, t + h) F (f, t )h

= 1h [1+ t, 1+ t + h ] [t,t + h ]


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xii INTRODUCTION

has norm

2/h and does not converge to a member of L2 (R) as h 0. Thelinear structure of the vector space L2 (R) is not helping in our quest to analyze 1

F .Even more fundamentally bothersome is the fact that the speed of the ow is

not locally bounded, i.e., the speed of the curves F (f, ) can become arbitrarilylarge on any neighborhood of M .(Here we are referring to the notion of speed dened technically above. The

speed of F (f, ) is not related to the rate the graph is translated on the R axiswhich is constantly 1. The metric is biased toward the structure of additionof functions in order to achieve a norm and is less sensitive to comparing howsimilar the graphs appear. Reread the denitions carefully so as not to be misledby initial intuition.)

This difficulty with translation is at the heart of many obstacles to answeringthe well-posedness of partial differential equations (PDEs), since translation isthe solution of

F t =

F x .

This is the simplest non-trivial partial differential equation and yet we alreadysee the unbounded property of some functional analysis operators rearing itshead. This warns us about the difficulties inherent in transporting the languageand intuition of continuous dynamics in nite dimensions to innite dimensionsor more general metric spaces. L2 is a beautiful, complete metric space whichis natural to consider as an environment for solving PDEs, but the pitfall men-tioned in this paragraph may lead us to widen our search to other metric spaces.

Example 3 Another basic ow on M := L2 (R) is vector space translation,G : L2 (R) R L2 (R) given by

G (f, t ) := f + tg

for any choice of gL2 (R). The evolution of the graph of Gt (f ) as t changes

is not quite as easy to visualize as Example 2; but since G respects the vector space structure, it is much tamer analytically. Verifying the ow properties is trivial. Continuity in particular follows immediately from the properties of the norm. In fact the speed is globally bounded by := g since

d (G (f, t ) , G (f, s )) = (f + tg) (f + sg) = |t s| g .G (, t ), like F (, t ) above, is again a family of isometries of M :

G (f 1, t )

G (f 2 , t ) = (f 1 + tg)

(f 2 + tg) = f 1

f 2 .

1 This difference quotient does, of course, converge to a difference of Dirac point distrib-utions t +1 t if we bother to dene the wider notion of a distribution in the linear dual.Admittedly were being overly critical on the value of linearity at this stage, but read on andnote for yourself why even the use of covectors wont simplify the analysis.


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0.2. EXAMPLE: FLOWS ON L2 (R) xiii

How do our two ows F and G from Examples 2 and 3 compare? How dothey interact on M , and what does this tell us about M ? Let us determine thereachable set for this pair of ows. The reachable set is an object of fundamentalconcern in the subject of control theory, which we take up in greater detail in3.5. Imagine we are running some process which allows us to apply either owF or G successively, at will, to an initial condition in our conguration spaceM . The reachable set starting from the initial point f is then dened as

RF,G (f ) := Gs n F t n Gs n 1 F t n 1 ...G s 1 F t 1 (f )M |s i , t iR, nN .Here we are dropping the composition parentheses, using Gs F t (f ) = G (F (f, t ) , s)to simplify notation; the general associativity of composition means the extraparentheses are unnecessary. So starting with the initial condition f M wecan steer our process in nite time to any conguration in RF,G (f ) M by judiciously applying F and G by various amounts s i and t i .

If RF,G (f ) is dense in M , then M is said to be controllable by F andG. For instance we could imagine M consists of the space of possible signals acircuit can generate in a looped line. G then represents adding a waveform inthe shape of the graph of g; and F would correspond to time lag as the signalnaturally cycles around the loop. The reachable set in this idealized scenariorepresents the possible signals that can be generated with our circuit.

As a rst inquiry into the nature of RF,G (f ) for our two types of translationon L2 (R), let us test whether F and G commute, i.e., does F (G (f, s ) , t ) =G (F (f, t ) , s )? If so the reachable set will be merely a two-dimensional subsetof the innite-dimensional space M = L2 (R), since any member collapses tothe simple representation

Gs n F t n Gs n 1 F t n 1 ...G s 1 F t 1 (f ) = Gs 1 + ... + s n F t 1 + ... + t n (f ) = Gs F t (f ) .

Perhaps surprisingly F and G are usually far from commutative, and by howmuch depends on the function g:

[F (G (f, s ) , t ) G (F (f, t ) , s)] (x) (3)= [f (x + t) + sg (x + t)] [f (x + t) + sg (x)] = s [g (x + t) g (x)] .

From the point of view of differentiable ows on a manifold, we would at leastexpect

d (F (G (f, t ) , t ) , G (F (f, t ) , t )) = O t2

and, in fact, continuing from line (3) we calculate

limt0F (G (f, t ) , t )

G (F (f, t ) , t )

t2 =dgdx

if g is differentiable. Following the ideas of geometric control theory, this breakin holonomy suggests the reachable set is more than two-dimensional. In factRF,G (f ) should be dense in the span of the set of all Lie brackets generated byF and G.


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xiv INTRODUCTION

This turns out to be exactly correct:

span dndxn

g nN RF,G (f ) (4)

where S denotes the topological closure of a set S M, and spanS denotesthe closed linear span of S in M . There are algorithms for steering any initialcondition to a member of the reachable set, and for many choices of gM , e.g.,g (x) := ex 2 , we nd that all of M is controllable. Continuing the applicationof this model to signal processing above, this means that for the correct choiceof g, any signal can be synthesized by alternately applying F and G. In thecourse of this book we will clarify the terminology and ideas surrounding theseclaims, culminating in 5.1 with Theorem 88.

One corollary is that low-frequency trigonometric series of the formN

n =1an eix/n

are dense in L2 [a, b] for any interval [a, b], Theorem 90. Heres a quick moti-

vation for this shocking fact: approximate f (x) N

n =1bn xn then notice xn =

in dndt n eitx t =0 . Now approximate the derivatives with nite differences (theformula is reviewed in Example 129). Example 95 gives the coefficients for ap-proximating x3 to arbitrary accuracy using just 3 low-frequency sine functions.

To achieve these results we generalize the Chow-Rashevsky Theorem to met-ric spaces: the closure of the reachable set is the closure of the integral manifoldto the distribution consisting of the set of all arc elds bracket-generated fromF and G, which gives line (4). The proof uses generalized versions of folia-tions (Chapter 3), Lie brackets (2.2), geometrical distributions (3.4), and anarithmetic of ows which works on a geometric as well as algebraic level (2.1,

Theorem 43). To give a rm footing for such complicated constructions us-ing only the abstract building blocks of metric spaces, we devote Chapter 1 tocarefully establishing the fundamental properties of existence, uniqueness andregularity of ows on M and Chapter 2 to the algebraic properties.

0.3 Example: ows on manifoldsThe ultimate source of inspiration for metric space generalization of geometri-cal and dynamical ideas is the theory of differentiable manifolds, in addition tobeing the colliery for our examples. The elaborate apparatus constructed to docalculus on a differentiable manifold is remarkably successful in extending tra-ditional calculus on Rn to a more general setting, indispensable in fundamental

areas of mathematics and physics. Much of this apparatus is ready to be furtherextended to metric spaces.Digesting the brief overview of differentiable manifolds in this section is not

necessary to digest the rest of the material in this book, but a familiarity withmanifold theory will allow you to anticipate all our results. This is an apologyto the beginner for how dense the next paragraph is. [23] or [12] or many other


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0.3. EXAMPLE: FLOWS ON MANIFOLDS xv

proper introductions expand the following paragraph to chapters; [1] gives anintroduction to innite-dimensional manifolds, or Banach manifolds. We makeno pretense toward rigor in this section, but promise to rectify this imprecisionin the presentation of the generalizations throughout the remainder of the text.We focus instead on the properties of manifolds which are naturally generalizedto metric spaces.

Example 4 Dene the torus

T 2 := S 1 S 1 = {(x mod2, y mod2) = ( x, y)mod2 |x, yR}where x mod2 is the remainder upon dividing x by 2. The torus T 2 is most easily geometrically visualized when embedded in R3 as a doughnut: First embed the circle S 1 in R3 via a map such as c (t) := (0 , 3 + sin t, cos t), then rotate the circle around the z-axis with the embedding S : T 2 R3

S (s, t ) :=cos s sin s 0sin s cos s 0

0 0 1

03 + sin t

cos t.

(This particular construction is easily extended to give more 2-dimensional man-

Figure 1: T 2 embedded in R3

ifolds such as found on pp. 43-47. )T 2 is an archetypical example of a manifold , which by denition is a set

which is locally at; this means every point xT 2 has a neighborhood U with

: U V a homeomorphism onto its image in the topological vector space V .For the torus V = R2 which makes it 2-dimensional, hence the superscript in T 2 . is called a chart for T 2 which gives local coordinates on the manifold. T 2is also a differentiable manifold, which means the charts match up nicely,which means for any two charts i : U i V for i = 1 , 2 the composition 2 11is a differentiable map from V to V wherever it is dened. The existence of


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xvi INTRODUCTION

these nicely-matched-up charts means any calculus done on V may be applied toT 2. Charts are easy to construct for T 2 ; the only difficulty arises near a point (x, y) T

2 when x = 0 or y = 0 , which the reader is encouraged to resolve.In Figure 1 you can see the locally at patches of R2 nicely matching up as grid lines to form the manifold, but this is an aberration amongst manifolds.Excepting the Klein bottle, no other compact 2-dimensional manifold has such perfectly aligned patches globally. Consider, e.g., a globe where longitude and latitude grid lines degenerate at the pole; results from topology prove any way you attempt to construct such a grid on a globe, will end with at least one point of degeneration [40].

Particularly important for our investigation of geometry and dynamics isthe concept of the tangent space of a manifold, roughly the space of all possibledirections in which you can move from a point within the manifold. The tangentspace may be dened in several equivalent ways; lets outline the most relevantfor our purposes. A tangent vector v at a point xM is an equivalence classof curves in the manifold c : ( c , c) M with c (0) = x under the equivalencerelation that c1 c2 if ( c1) (0) = ( c2) (0) for any chart , i.e., c1 andc2 are differentiable and tangent to each other at x. This equivalence relationdistills the idea of a direction with magnitude v located at the position xinto an abstract mathematical object, represented by an explicit, constructibleobject c (0). The tangent space at x is the set of all equivalence classes underthis relation, denoted T x M . The tangent bundle TM is the collection of all tangent spaces, i.e., the disjoint union TM :=

xM T x M , representing all

possible directions of motion (x, v ). A vector eld is a map f : M TM withf (x)T x M , and represents a rule for motion on the manifold. A solution toa vector eld is a curve : (, ) M which is tangent to the vector eld at allpoints, i.e., a curve which follows the rule. More concretely, if the translationmap s : R R is given by s (t) := s + t then s : ( s, s) M has s f ( (s)) for all s(, ). In other words follows the rules of motion of f through M . Assuming 0(, ) we say (0)M is the initial conditionof the solution, the place where the motion begins. By the FundamentalTheorem of ODEs (Appendix B) applied to charts, we always have solutions toa smooth vector eld, and we can combine them to give a local ow. Conversely,any differentiable ow on a manifold is generated by a vector eld.

Example 5 A simple ow that spirals around the torus F : T 2 R T 2 is dened by F t (x, y ) = ( x + t, y + at )mod2 . If a is a rational number then the path F (x, 0) (R) closes and is homeomorphic to the circle S 1 (Figure 2 ). These paths are evocatively described as toral helices. The different paths, starting

from different points (x, 0), partition T 2. This partition is an example of a 1-dimensional foliation of the torus. Two more foliations perpendicular to each

other are illustrated by grid lines in Figure 1 above.When a is not a rational number the path F (x, 0) (R) does not close on itself

and is homeomorphic to R, as a dense subset of T 2 (Figure 3 ). Still there are an innite number of disjoint paths, starting from different points (x, 0), which


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0.3. EXAMPLE: FLOWS ON MANIFOLDS xvii

Figure 2: Rational ow-path

again foliate T 2 .

Figure 3: Irrational ow-path

The solutions to the vector eld f (x, y ) := (1 , a) generate the ow F . Here the number (1, a) really represents the curve

c (t) := [( x, y ) + t (1, a)]mod2T 2

which itself is a representative of an equivalence class of curves under tangency and so [c]T (x,y ) T

2 . Sometimes its easier to just construct the ow than to think about the vector elds; but vector elds are generally considered pri-mary, and often have great descriptive power, giving a link between algebra and


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xviii INTRODUCTION

geometry. E.g., the vector eld illustrated here

V (x, y ) := (cos y, sin x)is smooth on T 2 since it matches up at 0 and 2. V is easily solved on the plane and transferred by charts (or S ) to the manifold. However, the ow paths of V do not foliate T 2 as before, since there are 4 xed points; instead this leads to a stratication since the paths are of different dimensionnamely 1 and 0.

Another inequivalent foliation of T 2 is given by the paths in Figure 4, consist-ing of two closed circles and an continuums worth of toral helices which accumu-late on the circles. More circles may be added, producing topologically distinct foliations. Essentially the nal twist we can add in foliating a 2-dimensional compact manifold is a Reeb component, illustrated in Figure 5. See [40] for an elementary classication of foliations on compact manifolds.

Higher-dimensional foliations of manifolds are vital to the study of geometryand dynamics. Examples of 2-dimensional foliations of a 3-dimensional spaceare illustrated on pages 42-47. Just as integral curves and 1-dimensional fo-liations are generated by vector elds or by 1-dimensional subbundles of thetangent bundle TM , surfaces and n-dimensional foliations are generated by ntransverse vector elds or by n-dimensional subbundles of TM called distribu-tions 2 . Not all distributions may be integrated to generate foliations, not even if they are smoothly dened (see Example 58). However, a simple condition calledinvolutivity characterizes the integrable distributionsthis characterization isreferred to as Frobenius Foliation Theorem.

To dene involutivity, we use the Lie bracket [f, g ] of two vector elds f and g, which is a new vector eld on M . The vector [f, g ] (x) is the tangency

equivalence class represented by the curve G t F t G t F t (x) where F andG are the respective ows of f and g. I.e., start at x M and move in an2 The term distribution is not to be confused with the several other mathematical con-

cepts that share its name. As a striking case of poor terminology, when studying dynamicson abstract function spaces three of these denitions may be needed in a single example:probability distributions, functionals, and subbundles, e.g, in Example 85.


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0.3. EXAMPLE: FLOWS ON MANIFOLDS xix

Figure 4: 4 leaves of another toral foliation are depicted: 2 circles and twopartially-complete squished toral helices. Everyone loves a Slinky.

Figure 5: Reeb component


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xx INTRODUCTION

approximate-parallelogram following F , then G, then F backwards, then Gbackwards. The little parallelogram almost returns to x, but t has innitespeed at t = 0 which cancels the naturally tendency of the parallelogram toclose, at least to order O (t), giving a curve with nite speed, Figure 6.Why

Figure 6: O (t) gap represents the new vector [f, g ]

do we use t? If we restrict our attention to x Rn and move > 0 ineach of the directions around the parallelogram in with the f and g vectoreld directions starting at x0 , then using Taylor series, we get a curve x ( ) =x0 + 2 gx f (x0) f x g (x0) + O 3 .

The bracket encapsulates a subtle difference between f and g which is crit-ical to appreciate. For example [f, g ] = 0 if and only if F and G commute,meaning the parallelogram closes perfectly. But there is much more. The Liebracket gives us fundamental geometrical information about the subbundle of TM generated by f and g, i.e., the distribution

( f, g ) := {span {f (x) , g (x)T x M }|xM }.The distribution ( f, g ) gives a plane at each point x and so is also called a planeeld. Frobenius Foliation Theorem says ( f, g ) foliates M into 2-dimensionalsurfaces ( leaves ) exactly when [f, g ] (x) ( f, g ) for all x M . Higher-dimensional foliations are determined by a straightforward generalization.

Frobenius Theorem has an important corollary for control theory, the Chow-Rashevsky Theorem, concerning the reachable set of a control system: If ( f, g )is involutive then the situation is simple, and the reachable set R f,g (x) is theleaf of the foliation through x; both sets consist of the set of all points in the set

of all piecewise differentiable paths containing x with derivatives being linearcombinations of f and g. If ( f, g ) is not involutive then [f, g ] is transverseto any surface tangent to f and g, so cycling through the motions of f and gaccording to the bracket denition sends us away from the tangent surface, andthus the reachable set is not as simple as in the involutive case. But there is asimple solution in this case as well. If ( f, g ) is not involutive, then we may form


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0.4. EXAMPLE: FLOWS ON A SPACE WITH NO LINEAR STRUCTURE xxi

the distribution [ f, g ] bracket-generated by f and g, consisting of the linearcombinations of f and g and all nitely iterated brackets, such as [[f, [f, g ]], f ].By denition [ f, g ] is involutive and so foliates M by Frobenius Theorem.The Chow-Rashevsky Theorem says the closure of the reachable set R f,g (x) isthe leaf of the foliation from the bracket-generated distribution [ f, g ] throughx. This is easy to believe now since iterated brackets of f and g are tangent tothe ows of some complicated composition of F and G. E.g.,

[f, [f, g ]] F 4 |t |G 4 |t |F 4 |t |G 4 |t | F t G 4 t F 4 t G 4 t F 4 t F t (x) .So the ow of any iterated bracket is in R f,g (x). This shows that the leaf is contained in the closure of the reachable set; the (less interesting) reverseinclusion follows from the Nagumo-Brzis invariant-set theorem, proven in 1.3.

Vector elds on a manifold are generalized to metric spaces with arc eldsas a special family of curves (cf. the technical description on page 3). Thedenition of Lie brackets (2.2) is essentially the same on metric spaces as givenabove for manifolds. But to dene distributions (3.4) using spans of arc elds,and also to dene involutivity, we need an arithmetic for ows on a metric spacebut metric spaces have no usable linear structure by denition. Surprisingly,though you cannot add points together in a metric space, you can add arc eldsin a natural way which faithfully generalizes the linear properties of vectorelds on manifolds: scalar multiplication is dened by changing the speed of the curves, and arc elds can be added simply by composing them (2.1). Thenglobal foliations on metric spaces follow with a new proof of Frobenius Theorem(Chapter 3). The Chow-Rashevsky Theorem is generalized in 3.5.

0.4 Example: ows on a space with no linearstructure

As a nal introductory example we consider a metric space which resists anynatural ascription of a linear structure, but still gives a fertile environment fordynamics.

Example 6 Let (Rn , d) be the usual n-dimensional Euclidean space. The met-ric space H (Rn ) is the set of all nonempty compact subsets of Rn and the Hausdorff distance is given by

dH (a, b) := max supx

ainf y

b {d (x, y)} , supy

ainf x

b {d (x, y)} .Using the simplifying notation d (x, a ) := inf

ya {d (x, y )}=: d (a, x ) for x Rn

and aRn , we have

dH (a, b) = supxa ;yb

{d (x, b) , d (y, a )}.


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xxii INTRODUCTION

H (Rn ) has several useful topological properties in common with Rn . It is separable, complete and even locally compact (separability is obvious by consid-ering nite subsets of Rn ; for completeness, see [8]; for local compactness, see [33, p. 183] ).

What makes this space interesting for modeling is that shapes of homoge-neous matter are merely points in this metric space. A circle, a rectangle, apentagram: all points in H R2 . A ball, a box, a cloud: points in H R3 .

Exercise 7 Find dH (a, b) when aH R2 is the unit coordinate box

a := {(x, y)|0 x 1, 0 y 1}and bH R

2 is the unit ball

b := (x, y )

|x2 + y2

1 .

Hint : Since a = b we cannot have dH (a, b) = 0 .

Figure 7: dH (ball, square ) =?

Exercise 8 Determine which points are in the ball BdH (0, 1)H R2 .

Hint : BdH (0, 1) = Bd (0, 1) and the word point is easily misinterpreted here.

As further motivation for the potential of this space, answer the question:

What is a curve in H (Rn ) ?

Looking at a black and white newspaper photo with a magnifying glass we see anite collection of black dots. This photograph may be thought of as a point inH R2 , the compact set representing the union of the black dots which forms


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0.5. CHAPTER OUTLINE xxiii

a closed and bounded subset of R2. Now if a black and white photograph is apoint in H R2 , a black and white lm clip is a curve in H R2 . These points of H R2 , the photographs or individual frames of the movie, move continuouslywith respect to the Hausdorff metric as time goes by (or at least approximatelycontinuously, since there are only a nite number of frames in a lm clip ). Colorcinema is a curve in H R3 .

The ability to describe the motion of complex patterns makes H (Rn ) a veryinteresting space. It is easy to imagine the motion and evolution of a homoge-neous material simply as a curve in this metric space: a moving cloud may becharacterized as a curve in H R3 , and a lightning stroke is a very fast curvein H R3 ; the growth of a bacteria colony in a petri dish and the evolution of asix-sided snowake growing from a tiny ice seed are both geometrically curvesin H R3 .

Very well then, H (Rn

) has a strong potential for describing all sorts of shape changes, but do we have any control on this profusion of informationwith which H (Rn ) presents us? How do we mathematically encapsulate motionor characterize forces on such a space? Can we generalize differential equations,somehow? Even then, could we stomach any calculations with this complicatedmetric? Happily, all of these questions have positive answers.

Lets construct some curves in H (Rn ).

Example 9 For x, yRn , let xy : [1, 1] Rn be the line dened by

xy (t) := (1 t) x + ty.For k functions f i : Rn Rn dene the arc eld X : H (Rn )[1, 1] H (Rn )by

X a (t) := xai=1 ,...,kxf i (x ) (t)

which describes curves from X a (0) = a to X a (1) = i=1 ,...,k f i (a) in H (Rn ). In

Chapter 7 this arc eld X is shown to generate a ow F on H (Rn ). When the f i are affine and contractive F t (a) converges to a unique xed point in H (Rn )as t , the convex hull of a fractal. Another example in 7.5 characterizes the reachable set of a control system as the limit of the ow of a similar arc eld.

0.5 Chapter outline

Part I: Theory In these chapters our will is bent to proving generalizationsof the basic theorems of dynamical systems and differential geometry.

Chapter 1: Flows The Fundamental Theorem of ODEs is generalized, prov-ing the well-posedness of arc elds, Theorem 12. This gives a meansfor generating ows on metric spaces. Global ows are guaranteed


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xxiv INTRODUCTION

when an arc eld satises the extra condition of linearly boundedspeed, Theorem 25. A xed point is guaranteed when the arc eld issuitably contractive, Theorem 31. An invariant-set theorem general-izing the Nagumo-Brzis Theorem is given with Theorem 33, whichis used later to piece together integral surfaces in a global foliationtheorem. Theorem 35 gives a condition analogous to a vanishing Liebracket which guarantees forward ows commute.

Chapter 2: Lie algebra on a metric space An arithmetic for arc elds is in-troduced which generalizes the algebraic structure of vector elds ona manifold. Theorem 43 elucidates which module properties generalarc elds enjoy. Then the Lie bracket is introduced and its algebraicproperties are explored. Theorem 52 shows how pull-back and push-forward operations are natural with respect to this new Lie algebra.

Chapter 3: Foliations Transverse arc elds generate geometric distributions.The Lie bracket is used to prove a local Frobenius theorem, showinginvolutive distributions have integral surfaces, Theorem 62. Theseintegral surfaces are pieced together to foliate metric spaces, culmi-nating in a global Frobenius theorem, Theorem 75. A corollary of this result is an application to control theory with Chows Theoremon a metric space, Theorem 78.

Part II: Examples

Chapter 4: Brackets on function spaces The Lie bracket and the FrobeniusTheorem are applied to simple ows on L2 (R) to make good on thepromises of 0.2. Various foliations of L2 and other function spacesare explored.

Chapter 5: Approximation with non-orthogonal families Applications of theresults of Chapter 4 give surprising new approximation methods us-ing non-orthogonal families of functions such as translations of aGaussian e(x+1 /n )2 |nN in 5.1 and low-frequency trigonomet-ric functions eix/n |nN in 5.2.

Chapter 6: More ows on function spaces PDEs are rewritten as arc eldsavoiding derivatives.

Chapter 7: Flows on H (Rn ) A continuous version of the discrete IFS fractalgenerator and other ows with novel dynamics are introduced.

Some sections are not logically dependent on others. The fastest tour of thehighpoints is

1.1 2.1 2.2 3.2 4 5


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0.6. PREREQUISITES xxv

0.6 Prerequisites

Technically the prerequisites for understanding this book are very basic; a singlesemester of undergraduate analysis which introduces the concept of a limit ina metric space is sufficient. Weve made efforts to keep the book self-containedand gently introduce each concept. Certainly, those with experience in thedifferentiable-manifold presentations of ows, Lie brackets and foliations willnd this generalized environment easy to apprehend. When released from thedetails of charts, atlases and coordinates, new students may likewise nd theseconcepts simpler to grasp.

Several proofs are extremely long. This is a good place to apologize, justifyourselves, and prepare the reader. This is an abstract subject with concreteclaims. We are a bit defensive, therefore, and feel the need to detail everypedestrian step exhaustively. Instead of relying on our readers mathematicaldexterity in this unfamiliar terrain, we spoiled the fun and printed out six-pageproofs. Instead of slogging through, line by line, it may be more productive foryou to read the proofs outline, then create one yourself.

0.7 Abridged version of the bookGeneralizations of the major ideas in dynamics and geometry can be fruitfullymade to metric spaces. As well as greater descriptive power, the extra generalitygives insight into classical questions on innite-dimensional spaces.

A vector eld on a manifold is recast as an arc eld X , that is, a set of curves on a metric space M , each curve representing a direction, i.e., X is acontinuous map X : M [1, 1] M such that for all xM , X (x, 0) = x.Tangency between two arc elds X and Y is given by the condition

d (X x (t) , Y x (t)) = o (t) .

If X satises the regularity conditions E1 and E2 (p. 1.1.1) on a completemetric space, then there exists a unique local ow tangent to X . If X haslinearly bounded speed, it generates a global ow.

An arithmetic for arc elds is given by X + Y and aX for aR dened by

(X + Y )t (x) := Y t X t (x)

and(aX )t (x) := X at (x) .

The Lie bracket [X, Y ] of two arc elds is given by

[X, Y ] (x, t ) := G t F t G t F t (x)

where F and G are the ows generated by X and Y .A distribution is a set of arc elds. A distribution is involutive if for

any X, Y , we have [X, Y ] . An involutive distribution has a unique


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xxvi INTRODUCTION

maximal integral surface through each point in M . The integral surfaces, piecedtogether, foliate M .

One application on M := L2 (R) shows the ows F t (f ) (x) := f (x + t)and Gt (f ) := f + tg bracket-generate an innite-dimensional distribution wheng (x) = ex 2 , and the reachable set is all of M . Similarly G and Z t (f ) (x) :=eixt f (x) have an innite-dimensional bracket-generated distribution and L2 ([a, b] , C)is controllable with G and Z , for any choice of interval [a, b]. Consequently se-

ries of GaussiansN

k=0ak e(x +1 /k )

2or low-frequency trig series

N

k= M ak eix/k may

be made arbitrarily close to any square integrable function.

0.8 Acknowledgements

This theory took more than 10 years to commit to paper, though I had assumedit could be hammered out in a few months. Its all Axel Boldts fault. Tomy constant irritation, he corrected countless mistakes and misunderstandings,which really slowed down the creative process. He also introduced me to severalbranches of mathematics, which distracted me from metric spaces, and mademe a more versatile mathematician. Thanks for screwing up my focus, pal.Michael Green was the mathematician who gave me the most extensive anduseful feedback on this manuscript. David Bleecker suggested I write this book,which was the strangest thing I had seen him do, so I took him seriously. Hehas been my greatest supporter in the development of these ideas.

Except for my wife, Karen. Often when authors thank their wives, I imaginea shrew who speeds the writing of a book by folding her arms and tapping herfeet at the doorway to the study. But Karen took an interest in all the ideasin this book, even the applications outside her eld of expertise. She was mybest sounding board, my best critic. And by introducing me to fatherhood thenguiding me for a year abroad in China, shes been my best teacher.


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Part I

Theory

1


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Chapter 1

Flows

Panta rhei . (Everything ows.)-Heraclitus, ca. 500 B.C.

The purpose of this chapter is to introduce a general method for producingows (dynamical systems) on a metric space. A ow may proceed forward andbackward in time F : M (, + ) M , or possibly only forward in timeF : M [0, + ) M as in the case of diffusion. We explore the generation of both types of ows and study some conditions which guarantee global existence,xed points and commutativity.

1.1 Generating ows with arc elds

This section follows the generation of ows on a manifold M from a vector eld:rst we nd solutions for each initial condition xM , then we piece togetherthe solutions with domain (, ) in a neighborhood of x to get a local ow,which are then continued to produce a global ow with domain (, ).

1.1.1 The fundamental theoremThe following denition is made in analogy with the representation of a vectoreld on a manifold as a family of curves, detailed in 0.3.

Denition 10 An arc eld on a metric space M is a continuous map X :M [1, 1] M with locally uniformly bounded speed, such that for all xM ,X (x, 0) = x.

Saying X has locally uniformly bounded speed means X (x, ) : [1, 1] M is Lipschitz, locally uniformly in x. Specically we have (x) := sup

s = t

d (X (x, s ) , X (x, t ))

|s t|< ,

3


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4 CHAPTER 1. FLOWS

(i.e., X (x, ) is Lipschitz ), and the function (x) is locally bounded, meaningthere exists r > 0 such that (x, r ) := sup { (y) |yB (x, r )}< .

A solution curve to X is a curve which is tangent to X throughout itsdomain, i.e., : (, ) M for some open interval (, )R such that foreach t(, )

limh0

d ( (t + h) , X ( (t) , h))h

= 0 , (1.1)

i.e., d ( (t + h) , X ( (t) , h)) = o (h).Arc elds are typically denoted with X , Y , or Z . The two independent

variables for arc elds, usually denoted by x and t, are often thought of asrepresenting space and time. We typically use x,y, and z for space variables,while r,s,t, and h ll the time variable slot. As with ows, the variables of anarc eld X will often migrate liberally between parentheses and subscripts

X (x, t ) = X x (t) = X t (x)

depending on which variable we wish to emphasize in a calculation.On Rn a vector eld which is Lipschitz continuous generates a local ow

constructed by Euler curves. An arc eld is a faithful analogy for a metricspace, and when it satises analogous regularity conditions (E1 and E2 detailedbelow), we will soon show Euler curves converge to a ow. To further theanalogy with vector elds on manifolds, an arc eld may be thought of as amap X : M AM where AM is the arc bundle , consisting of the set of allLipschitz continuous arcs, and we require X (x) (0) = x.

The initial condition of is the point x = (0)

M . Notationally we usex to mean the solution with initial condition x. We say x : (x , x ) M is the unique solution to X with initial condition x if for any other solution

x : ( x , x ) M also having initial condition x, we have ( x , x )( x , x )and x = x |( x , x ) (i.e., x is the unique maximal solution curve ).We will prove below that on a locally complete metric space the next twoconditions guarantee the arc eld problem is well posed , i.e., there exists aunique solution from any initial condition xM (Theorem 12).

Condition E1: For each x0M , there exist positive constants r, and suchthat for all x, yB (x0, r ) and t(, )d (X

t(x) , X

t(y))

d (x, y) (1 +

|t

|)

Condition E2: For each x0M , there exist positive constants r, and suchthat for all xB (x0 , r ) and s(, ) and any t with |t| |s|d (X s + t (x) , X t (X s (x))) |st | .


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1.1. GENERATING FLOWS WITH ARC FIELDS 5

These conditions may be restated as saying

d (X t (x) , X t (y))d (x, y ) 1 O (|st |)

andd (X s + t (x) , X t (X s (x))) = O (|st |)

for |t| |s| as s 0, locally uniformly in x. Innitesimally these conditionshave E1 limiting the spread of X , and E2 restraining X to be ow-like (Figure1.1).

Figure 1.1: E1 and E2 are continuity conditions on X which ensure some geo-metric regularity using only the metric.

Example 11 A Banach space (M, ) is a complete normed vector space (e.g., Rn with Euclidean norm ). A Banach space is an example of a metric space (M, d ) where the metric is dened by d (u, v) := u v . A vector eld on a Banach space M is a map f : M M . A solution to a vector eld f with initial condition x is a curve x : (, ) M dened on an open interval (, ) R containing 0 such that x (0) = x and x (t) = f (x (t)) for all t (, ). The Fundamental Theorem of ODEs (detailed in Appendix B ) guarantees unique solutions for any locally Lipschitz vector eld f . With a few tricks, most differential equations can be represented as vector elds on a suitably abstract space.

Every Lipschitz vector eld f : M M naturally gives rise to an arc eld X (x, t ) := x + tf (x) on M , and it is easy to check X satises E1 and E2 :Calculating

d (X t (x) , X t (y)) = X t (x) X t (y) x y + |t| f (x) f (y) (1 + |t|K f ) x y

where K f is the local Lipschitz constant for f , so X := K f gives Condition


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6 CHAPTER 1. FLOWS

E1.

d (X s + t (x) , X t (X s (x)))= x + ( s + t) f (x) [X s (x) + tf ((X s (x)))] = tf (x) tf (X s (x)) |t|K f x [x + sf (x)] |st |K 2f x

so X := K 2f x . Further the solutions to the arc eld are precisely the solutions to the vector eld guaranteed by the fundamental theorem since

d (t + h) , X (t ) (h) = |h| (t + h) (t)

h f ( (t)) = o (h) (t) = f ( (t)) .

Therefore Theorem 12, below, is a generalization of the classical Fundamental Theorem of ODEs (given in Appendix B ). Similarly, Lipschitz vector elds on a Banach manifold (a manifold whose charts map to a Banach space; if f is locally Lipschitz in one chart, it is in any and on the manifold with any compatible metric ) give arc elds which satisfy E1 and E2.

The basic iterative trick for proving ODEs are well-posed on Rn , or moregenerally on a Banach space, applies just as well for arc elds on general metricspaces. For economy of description we use round brackets in the superscript,f ( i ) , to denote the composition of a map f : M M with itself i times. So, forexample,

X ( i )t/ 2n (x) = X t/ 2n X t/ 2n ... X t/ 2n

i compositions

(x) .

Then given x M and a positive integer n, we may dene the n-th Eulercurve n : (2n , 2n ) M for X starting at x as n (t) := X

(2 n )t/ 2n (x) (1.2)

for nN such that 2n > |t|. Taking n generates a solution to X in thefollowing fundamental result.

Theorem 12 Let X be an arc eld satisfying E1 and E2 on a locally complete metric space M . Then given any point x M , there exists a unique solution x : (x , x ) M with initial condition x (0) = x for some x < 0 < x R{, }.

Proof of Existence of Solutions. We will showlim

n n (t) = limn

X (2n )

t/ 2n (x)

exists for each t sufficiently close to 0 and dene x (t) as this limit. Then x (t)will be shown to be tangent to X at t = 0 . The elaborate chain of elementary


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calculations checking these two facts becomes convoluted, but the inspirationguiding us is sketched simply enough in Figures 2.2 and 2.3. We then establishx (s + t) = x (s ) (t) which shows x is tangent to X at all t in its domainby the previous result. Uniqueness of solutions is elaborated and veried inRemark 17, below.

First we show that for sufficiently small c > 0 the image of the Euler curves n ([c, c]) must remain bounded for all n. This is intuitively true because thearc eld X from which the Euler curve is constructed has locally bounded speed < , so successively following 2n compositions of X for small time t/ 2n doesnot allow us to travel further than |t| distance. This is exactly correct, but weneed to demonstrate how we can achieve this bound using only the metric. Weexhaust the rest of this voluminous paragraph with the tedious details. Supposer > 0 is chosen so (x, r ) < . If (x, r ) = 0 , then (t) := x denes a solutioncurve and there is nothing to prove. Thus, assume (x, r ) > 0, and let

c := r/ (x, r ) .

We assume hereafter that t is restricted to |t| < c and |t| < 1, guaranteeing theEuler curve n (t) is well dened. In this case we claim n (t) B (x, r ): thetriangle inequality gives

d (x, n (t)) 2n

k=1d X (k1)t/ 2n (x) , X

(k )t/ 2n (x)

where X (0)t/ 2n (x) = x by denition.

d X (k1)t/ 2n (x) , X (k)t/ 2n (x) = d y, X t/ 2n (y) (y) |t|/ 2n for each k

where y := X (k1)t/ 2n (x). So if yB (x, r ) then (y) (x, r ) and inductionallows us to conclude

d ( n (t) , x) (x, r ) |t| < r .

Next we additionally assume the above r > 0 is chosen small enough that

and from Conditions E1 and E2 hold uniformly on B (x, r ) and for conve-nience, that , > 1. We may further assume the closure B (x, r ) is a completemetric subspace of M by again taking r to be smaller if need be. In this carefullychosen neighborhood we will now show the Euler curves converge by proving nis Cauchy. (If M were locally compact, Arzela-Ascoli would allow us to bypassthis one page verication.)


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8 CHAPTER 1. FLOWS

Figure 2.2: To prove the Euler curves areCauchy, apply E1 and E2 repeatedly toestimate the distance between n (t) and n +1 (t) tracking back to n (0) = x0 = n +1 (0) .

Figure 2.3: To prove tangency applyE1 and E2 to estimate the distancebetween X t (x0) and n (t) x (t) .

Consider

d n (t) , n +1 (t) = d X (2 n )t/ 2n (x) , X

(2n +1 )t/ 2n +1 (x)

d X (2 n )t/ 2n (x) , X

(2)t/ 2n +1 X

(2 n 1)t/ 2n (x) + d X (2)t/ 2n +1 X

(2 n 1)t/ 2n (x) , X (2n +1 )t/ 2n +1 (x)

The rst term is approximated by

d X (2n )

t/ 2n (x) , X (2)t/ 2n +1 X

(2 n 1)t/ 2n (x) = d X 2t/ 2n +1 X (2 n 1)t/ 2n (x) , X

(2)t/ 2n +1 X

(2 n 1)t/ 2n (x)

= d X 2t/ 2n +1 (y) , X (2)t/ 2n +1 (y)

t2n +1

2

for y := X (2n 1)t/ 2n (x) using Condition E2, while the second term is approximated

by

d X (2)t/ 2n +1 X (2 n 1)t/ 2n (x) , X

(2n +1 )t/ 2n +1 (x)

= d X (2)t/ 2n +1 X (2 n 1)t/ 2n (x) , X

(2)t/ 2n +1 X

(2n +1 2)t/ 2n +1 (x)

d X (2n

1)t/ 2n (x) , X (2n +1

2)t/ 2n +1 (x) 1 + |t|2n +1

2

using Condition E1 twice. Such calculations will now be performed frequentlyand without comment for the rest of the proof; usually when a new orerupts, the triangle inequality and Condition E1 or E2 have been used.


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Inserting these last two estimates and iterating we have

d X (2n

)t/ 2n (x) , X (2n +1

)t/ 2n +1 (x)

d X (2 n 1)t/ 2n (x) , X

(2n +1 2)t/ 2n +1 (x) 1 + |t|2n +1

2

+t

2n +12

d X (2 n 2)t/ 2n (x) , X

(2n +1 22)t/ 2n +1 (x) 1 + |t|2n +1

22

+ 1 + |t|2n +1

2 t

2n +12

+t

2n +12

d X (2 n 2n )t/ 2n (x) , X

(2n +1 22n )t/ 2n +1 (x) 1 + |t|2n +1

22n

+2n 1k=0 1 + |

t

|2n +1 2k t

2n +12

= 0 +t

2n +12 2n 1

k=01 + |t|

2n +1

2k

(geometric series )=

t2n +1

2 1 + |t |2n +1 2n +1

11 + |t |2n +1

2

1

=t

2n +12 1 + |t |2n +1

2n +1

1|t |2n + |

t |2n +1 2

t2n +1

2 e|t | 1|t |2n

|t|2n +1

e|t | 1

Then for m < n

d ( m (t) , n (t)) n 1k= m

d k (t) , k+1 (t) =n 1k= m

d X (2k )

t/ 2k (x) , X (2k +1 )t/ 2k +1 (x)

n 1k= m

|t|2k+1

e|t | 1 |t|

e|t | 1

2(m +1) k=0

2k = |t|e|t | 1

2m

and we see n (t) is uniformly Cauchy on the interval |t| < c in the completemetric space B (x, r ). By the bound on speed, the curves n (t) are uniformlycontinuous in t and so they converge to a (continuous) curve, denoted

x (t) := limn n (t) .

Let us now check x is tangent to X , rst at t = 0 . Notice

d (x (t) , X x (t)) d (x (t) , n (t)) + d ( n (t) , X x (t)) .The rst summand is easily controlled. For the second summand consider thefact that for any t[1, 1] and nN we have

d X t (x) , X (n )t/n (x) e|t | t2 (1.3)


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10 CHAPTER 1. FLOWS

which holds since

d X t (x) , X (n )t/n (x) n 1k=0 d X (n k)t/n X kt/n (x) , X (n (k+1))t/n X (k+1) t/n (x)

n 1k=0

1 + |t|n

(n (k+1))

ktn

2

e|t |t2 .

Replacing the n in (1.3) with 2n , the bound is undisturbed, and we have

d (x (t) , X x (t)) d (x (t) , n (t)) + e|t |t2.Letting n gives

d (x (t) , X x (t)) e|t |t2 = O t2 (1.4)locally uniformly in x.

Next we show x is locally 2nd-order tangent to X for all t. This will bedone if we show x (s + t) = x (s ) (t) because in that case

d x (s + t) , X x (s ) (t) = d x (s ) (t) , X x (s ) (t) = O t2

this last equality having been established by line (1.4). Using (1.3) we have

d X t (x) , X (n )t/n (y) d (x, y) + t2 e|t | (1.5)since

d X t (x) , X (n )t/n (y) d X t (x) , X

(n )t/n (x) + d X

(n )t/n (x) , X

(n )t/n (y)

t2 e|t | + 1 + |t|n

n

d (x, y ) d (x, y ) + t2 e|t |.

Next if k divides j then using (1.5) we have

d X ( j )t/j (x) , X ( j/k )kt/j (x) e|t |t2 k/j (1.6)

since

d X ( j )t/j (x) , X ( j/k )kt/j (x) = d X

(k )t/j X

(k[j/k 1])t/j (x) , X kt/j X ( j/k 1)kt/j (x)

d X (k [j/k 1])t/j (x) , X ( j/k 1)kt/j (x) + kt j2

e|kt/j | ...

... d X (0)t/j (x) , X

(0)kt/j (x) +

ktj

2e|kt/j |+ e|kt/j | + ktj

2e|kt/j |

+ ... ktj2 e|

k


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where the sum is taken j/k times and since d X (0)t/j (x) , X (0)kt/j (x) = 0 the

above is

...kt j

2

e|kt/j |+ e|kt/j | + kt j

2

e|kt/j | + ... kt j

2

e|kt/j |

=kt j

2

= t2 e|t | k j

.

(1.6) is useful because it gives us

limj

X ( j )t/j (x) = limj X (j/k )kt/j (x)

or better putlim

j X (kj )t/j (x) = limj

X ( j )kt/j (x) (1.7)

where k can be any function of j and t as long as k/j 0 as j .For each n N, choose i (n) N such that n/i (n) 0 as n (forexample, choose i (n) = n2 ). and let j (i, n ) , k (i, n )N be chosen so j

s + t2i

s2n

< |s + t|2i

and ks + t

2i t

2n< |s + t|

2i

so

js + t

2i+ k

s + t

2i s

2n+

t

2n< 2 |s + t|

2iwhich implies ( j + k) 2in < 2

so j + k = 2 in + (n)

where | (n)| < 2. Thereforex (s + t) = limn

X (2n )

(s + t ) / 2n (x) = limiX (

2i )s + t2 i

(x) = limi

X (2n [j + k])

s + t2 i

(x)

= limi

X (2n j )

s + t2 i

X (2n k)

s + t2 i

(x) and using (1.7) this is

= limi

X (2n )

j s + t2 i

X (2n )

k s + t2 i

(x) = limn

X (2n )

t/ 2n X (2 n )s/ 2n (x)

= limn

X (2n )

t/ 2n limn X (2

n )s/ 2n (x) = x (s ) (t) .

This completes the proof that solutions exist which are locally uniformly 2nd-order tangent to X . The proof of uniqueness follows from Theorem 16 below;see Remark 17.


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12 CHAPTER 1. FLOWS

Remark 13 Theorem 12 has a simple corollary showing the well-posedness of time-dependent dynamics following the exact same idea for time-dependent vec-tor elds on a manifold. Simply consider a time-independent arc eld on M R,namely ((x, t ) , h) (X x,t (h) , t + h) in M R, and project solutions onto the M factor.

2nd-order differential equations can be rewritten with 2nd-order vector elds.A 2nd-order arc eld is a straightforward generalization with well-posedness a simple corollary of Theorem 12 (see [16] for details ).

With a little extra effort Theorem 12 and those which follow are true in evengreater generality, and the reader is encouraged to study the work in, e.g., [52],[7] and [18]. **check on the status of columbo and corlis new work**. Butin the examples throughout this book the stronger conditions E1 and E2 aresatised and are Easier to use.

The above proof actually gives a result stronger than the statement of thetheorem which will be frequently useful:

Corollary 14 Assuming E1 and E2, the solutions are locally uniformly 2nd-order tangent to X in the variable x, i.e.,

d (X x (t) , x (t)) = O t2

locally uniformly for xM ; i.e., for each x0M there exist positive constants r,,T > 0 such that for all xB (x0 , r )

d (X x (t) , x (t)) t2T whenever |t| < .

Proof. This was established at line (1.4).Denote local uniform tangency of two arc elds X and Y by X Y andlocal uniform 2nd-order tangency by X Y . It is easy to check and areequivalence relations. E.g., transitivity follows from the triangle inequality:

d (X t (x) , Z t (x)) d (X t (x) , Y t (x)) + d (Y t (x) , Z t (x)) .We use the symbols and in many contexts in this monograph (particularly3.4), and always with an associated local-uniform-tangency property.

Further, the proof of Theorem 12 gives us another useful fact we will subse-quently need :

Corollary 15 Assuming E1 and E2, the solutions are tangent uniformly over all arc elds X which satisfy E1 and E2 for specied and .

Proof. This was also established at line (1.4).Also notice the proof used only the weaker property s = t and not the more

general |t| |s| from Condition E2 to prove the Euler curves are Cauchy. Thefull assumption was used to prove the solution is tangent to the arc eld.


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Theorem 16 Let x : ( x , x ) M and y : y , y M be two solu-tions to an arc eld X which satises E1. Assume ( x , x )

y , y

I for some interval I containing 0, and assume E 1 holds uniformly with on a set containing

{x (t) |tI }{y (t) |tI }.Then

d (x (t) , y (t)) e|t |d (x, y) for all tI .Proof. We check t 0, the case t < 0 being similar. Let

g (t) = et d (x (t) , y (t)) .

For h 0, we haveg (t + h)

g (t)

= e( t + h ) d (x (t + h) , y (t + h)) et d (x (t) , y (t))= e( t + h ) (d (X h (x (t)) , X h (y (t))) + o (h)) et d (x (t) , y (t)) et eh d (x (t) , y (t)) (1 + h) et d (x (t) , y (t)) + o (h)= eh (1 + h) 1 et d (x (t) , y (t)) + o (h)= o (h) et d (x (t) , y (t)) + o (h) = o (h) (g (t) + 1) .

Hence, the upper forward derivative of g (t) is nonpositive; i.e.,

D + g (t) := limh0+

g (t + h) g (t)h 0.

Consequently, g (t)

g (0) or

d (x (t) , y (t)) et d (x (0) , y (0)) = et d (x, y ) .

Theorem 16 says solutions locally diverge at most exponentially, which is themost useful result we have for proving regularity of ows. When I is compact

{x (t) |tI }{y (t) |tI }is compact since x is continuous, and so it is often easy to nd a uniform bound for E1 on the set.

Remark 17 Uniqueness of solutions in Theorem 12 has the same meaning as

in classical ODE theory :(1.) Any two solutions 1x : 1x , 1x M and 2x : 2x , 2x M with

initial condition x has 1x (t) = 2x (t) for all t1x ,

1x 1x , 1x and (2.) There exists a solution x : (x , x ) M with maximal domain,meaning any other solution x : (x , x ) M has in the sense that for any (x , x )(x , x ).


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14 CHAPTER 1. FLOWS

Choosing x = y in Theorem 16 establishes (1.) for a small interval containing the origin. The exact same extension argument as in ODEs then establishes (1.) and (2.) fully (cf. practically any text introducing ODEs, e.g., [39] ). The maximal interval (x , x ) described in (2.) is the union of the domains of all solutions with initial condition x.

Example 18 **Good spot for the non-unique solutions example x = x. This example indicates how E1 and E2 cannot be weakened too much if we want toguarantee a general well-posedness result.**

Remark 19 Theorem 16 gives uniqueness of solutions for any arc eld which satises E1 alone. E2 is only used to prove general existence, but E2 is typi-cally the more difficult condition to verify, so if we can verify solutions exist in

some other manner (perhaps directly calculating the limit of Euler curves, as in Example 100 ) E1 is sufficient.

Theorem 16 also gives an easy proof of a very general Nagumo-type invariant-set theorem, Theorem 33 below in 3.4.

Notice in the proof of Theorem 12 the Euler curves were dened with nodesspaced at a distance of t/ 2n . This was for convenience. The simpler expression

limn

X (n )t/n (x) = x (t) (1.8)

may also be veried, but we wont present the more tedious analysis.

Yet a third denition of Euler curves for any real number r > 0 is common:for i, n N dene

r,n (t) :=X ( tir 2 n ) X

( i )r 2 n (x) i r 2n t (i + 1) r 2n

X ( t + ir 2 n ) X ( i )

r 2 n (x) (i + 1) r 2n t i r 2n .

This concatenation of arcs is more complicated notationally, but more intuitivelycompelling, and is in introductory texts on differential equations. Again r,n x as n as was proven in [18] with r = 1 to verify well-posedness undercommensurate conditions. Notice

t,n (t) = X (2 n )t/ 2n (x)

since t = 2 n t2n .1.1.2 Local owsFrom now on (x , x ) will denote the maximal domain with initial condition x.


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Corollary 20 Assume the conditions of Theorem 12 and let s (x , x ).Then x (s ) = x

s and x (s ) = x

s. Thus t

x (s ) , x (s ) if and only if t + s(

x , x ), and then we have x (s ) (t) = x (s + t).

Dening W M R by W : = {(x, t )M R|t(x , x )} and F : W M by F (x, t ) := x (t) (1.9)

Then

(i) M {0}W and F (x, 0) = x for all xM (identity at 0 property )(ii ) F (t, F (s, x )) = F (t + s, x ) (1-parameter local group property )(iii ) For each ( xed ) xM , F (x, ) : ( x , x ) M is the maximal solution x to X .

The map F is called the local ow of X .Compare Condition E2 with (ii ) above to see why an arc eld might be

described as a pre-ow.Theorem 16 says if F is the local ow of an arc eld X which satises

Condition E1 with uniform constant X then

d (F t (x) , F t (y)) eX |t |d (x, y ) . (1.10)Thus F (x, t ) is continuous in x. Notice eX |t | = 1+ X |t|+ O t2 and compareCondition E1 with line (1.10) to see why E1 may be thought of as a local linearityproperty for X , needed for the continuity of F . Now lets check continuity inthe other variable, t:

Lemma 21 Suppose c > 0 and : (c, c) X is a solution curve of X .Assume the speed of X is bounded by [0, ) on ((c, c)) . Then the speed of is also bounded by .Proof. First let t 0. For c t0 t0 + t < c, let

f (t) := d ( (t0 + t) , (t0)) tSince f (0) = 0 we wish to show D + f (t) 0, since then f (t) 0 and we willthen know

d ( (t0 + t) , (t0)) tas desired.

f (t + h) f (t) = d ( (t0 + t + h) , (t0)) d ( (t0 + t) , (t0)) h

d ( (t0 + t + h) , (t0 + t)) h d ( (t0 + t + h) , X h ( (t0 + t))) + d (X h ( (t0 + t)) , (t0 + t)) h= o (h) + d (X h ( (t0 + t)) , X 0 ( (t0 + t))) h o (h) + h h = o (h) .

Checking d ( (t0 + t) , (t0 )) |t| for t < 0 is similar, mutatis mutandis .


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16 CHAPTER 1. FLOWS

Theorem 22 For F and W as above (1.9 ) we have W open in M R and F continuous on W .Proof. Continuity is easy to check using Theorem 16 and Lemma 21 on

the separate variables, once weve established a proper environment on whichtheir assumptions are satised. So we rst check W is open by showing forany (x0 , t0) W there is a neighborhood V of x0 and > 0 such that V (t0 , t0 + )W . Dene x1 := x0 (t0). Since X has locally bounded speedthere exists r > 0 such that := (x1 , r ) < , and so for any xB (x1, r/ 2)we have x (t)B (x1 , r ) for |t| < r2 . Consequently B x1 , r2 r2 , r2 W .

Now the rest of the proof follows the idea that there is a small enoughneighborhood V of x0 such that F (V, t0)B (x1 , r/ 2) by Theorem 16 whichguarantees V

t0

r

2 , t0 +r

2

W since

F F (V, t0 ) , r2 , r2 = F V, t0 r2 , t 0 + r2by the local group property. Theorem 16 requires only there be a set on whichE1 is satised uniformly by some > 0. Then V := B x0, et 0 r4 is suffi-cient. Now to show the set for Theorem 16 exists. Notice 0, t 0 + r2 is compact

and so its continuous image x 0 0, t0 + r2 M is compact. For each t 0, t0 + r2 there is a ball B (x 0 (t) , r t )M with r t > 0 on which Condition

E1 is satised with t . These neighborhoods cover x 0 0, t 0 + r2 , so there is

a nite subcover {B (x 0 (t i ) , r t i ) |i = 1 ,...,n }. Let := max {i |i = 1 ,...,n },let U := ni=1 B (x 0 (t i ) , r t i ) and let M \U denote the set complement. Thefunction f : 0, t 0 + r2 R dened by f (t) := d (x 0 (t) , M \U ) is positiveand continuous on a compact domain and so has a minimum m > 0. There-fore any y x 0 0, t0 +

r2 has a neighborhood ball B (y, m ) U and

therefore any solution curve which stays within a distance of m of the pathx 0 0, t 0 + r2 will have a uniform satisfying E1. Therefore Theorem 16

applies and we can choose V := B x0 , et 0 r4 as explained above, giving(x0 , t0)V t0 r2 , t 0 + r2 W and W is open. (In fact we have provenV

0, t0 + r

2

W .)Now proving continuity is easy. Since X is an arc eld, it has locally bounded

speed and there exists r > 0 and a local bound on speed := (x 0 (t0) , r ) < for X y for all yB (x (t0 ) , r ), in particular Lemma 21 requires the speed of x 0 (t) be bounded by for all t with |t t0| < r2 . Using Theorem 16 (on theset constructed in the previous paragraph for which is uniform) and Lemma


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21, as (x, t ) (x0 , t0) we haved (F (x, t ) , F (x0 , t0))

= d (x (t) , x0 (t0)) d (x (t) , x 0 (t)) + d (x0 (t0) , x 0 (t)) e( |t 0 |+1) d (x0 , x) + (x (t0) , r ) |t0 t| 0.

For xed t it is clear F t is a local lipeomorphism, when dened, by Theorem16.

1.1.3 Global owsWe now investigate conditions which guarantee local ows are in fact global,i.e., (x , x ) = R for all xM . To achieve this, we mimic ODE theory.

Example 23 Consider the classic elementary example of quadratic speed growth x = f (x)

where f : R R is the (locally Lipschitz ) vector eld given by f (x) = x2 which has solutions x (t) :=

x01 tx 0

so that when the initial condition is x (0) = x0 =

0, the solutions blow up at time t = 1 /x 0 . The vector eld f (x) = x2 grows too quickly as solutions x grow, sending x to in nite time.

To guarantee globally dened ows, rst the space cannot have holes, i.e.,M must be complete. Secondly we must limit the magnitude of the vector eldto prevent the situation in Example 23, which inspires the following

Denition 24 An arc eld X on a metric space M is said to have linear speed growth if there is a point xM and positive constants c1 and c2 such that for all r > 0 (x, r ) c1 r + c2 , (1.11)

where (x, r ) is the local bound on speed given in Denition 10.

If y is any other point in X then B (y, r )B (x, d (x, y ) + r ) . Thus,

(y, r ) (x, d (x, y ) + r ) c1 (x) (d (x, y ) + r ) + c2 (x)= c1 (x) r + ( c1 (x) d (x, y ) + c2 (x)) .

Hence, if the relation (1.11) holds for a point x, then for any other yX wealso have

(y, r ) c1 (y) r + c2 (y) , (1.12)where c1 (y) = c1 (x) and c2 (y) = c1 (x) d (x, y ) + c2 (x).Theorem 25 Let X be an arc eld on a complete metric space M , which sat-ises E1 and E2 and has linear speed growth. Then F has domain W = M R,i.e., F is a ow .


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18 CHAPTER 1. FLOWS

Proof. A similar proof in this context of metric spaces appears in [18]. Mostother proofs on manifolds can be easily transferred to our current situation.

Assume t 0 (the case t < 0 being similar). Then for any partition 0 =t1 < t 2 < ... < t n +1 = t of [0, t ] we haved (x (t) , x) d (x (t) , x (0))

n

i=1d (x (t i ) , x (t i+1 ))

n

i=1 (x (s i )) |t i+1 t i |

for some choice of s i[t i , t i+1 ] which leads to

d (x (t) , x) t0 (x (s)) ds t0 c1 (x) d (x (s) , x) + c2 (x) ds.In other words, for f (t) := d (x (t) , x) we wish to use the inequality

f (t)

t

0

c1f (s) + c2ds (1.13)

to bound f . In fact (1.13) gives

D + f (t) := limh0+

f (t + h) f (t)h c1f (t) + c2 .

As motivation, the solution of x (t) = c1x (t) + c2 satisfying x (0) = 0 isx (t) = c2c1 (e

c1 t 1). Since we expect f (t) x (t) and f (0) = 0 , we expect f to grow at most exponentially. Then assuming the domain (x , x ) has x < we would have by continuity and the boundedness of f that x can be continuedto have x (x )M . But then the fundamental theorem allows us to continuex beyond x giving us the contradiction.

A ow is sometimes called a full ow, or a global ow, or a completeow to distinguish it from a local ow. Since local ows are continuousandcontinuity is a local propertyfull ows are continuous.

Example 26 The support of an arc eld X is the closure of the set S :=

{xM |X m 0}. Here 0 is the constant arc eld 0x (t) = x. Assuming E1 and E2 on a locally complete space M , it is easy to see that when the support of X is compact, the ow F is complete; in particular if M is compact all such X give complete ows.

Example 27 Every local ow on a metric space is generated by an arc eld.Any local ow F gives rise to an arc eld F : M [1, 1] M dened by

F (x, t ) :=F (x, t ) if t

x2 ,

x2

F x, x2 if t1, x2F x, x2 if t

x2 , 1 .

The issue here is that F , being a local ow, may have [1, 1] (x , x ), sowe have to be careful at the endpoints. Clearly the local ow generated by F is F . Since all our concerns with arc elds are local, we will never focus on t /

x2 ,

x2 and henceforth we will not notationally distinguish between F and

F as arc elds.


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1.2. FORWARD FLOWS AND FIXED POINTS 19

With this identication of ows being arc elds (but not usually vice -versa )we may simplify Corollary 14 to:

Corollary 28 X F if X satises E1 and E2.

Examples relevant to this chapter occur in Chapter 7 and Example 100.

1.2 Forward ows and xed pointsIn many applications the solution of a differential equation, or vector eld is notdened for t < 0. For example, diffusion phenomena is usually only tractableforward in time. In this case we work with forward ows (also called semi-ows , or in the context of operators on Banach spaces, semi-groups ). We listhere the minor modications to the above theory for this more general situation,then prove a simple xed point theorem. We dont bother to stress much newforward-specic terminology, as it should be clear from context whether we meanforward or bidirectional in any examples.

Change the domain of arcs on M from c : [1, 1] M to c : [0, 1] M and similarly replace [1, 1] with [0, 1] everywhere it occurs, e.g., (forward) arcelds are dened as maps X : M [0, 1] M . Solutions x : [0, x ) M areforward tangent to X dened by

limh0+

d ( (t + h) , X ( (t) , h))h

= 0 ,

i.e., t and h are restricted to positive values. We explicitly spell out the minorchanges to Conditions E1 and E2 since a new possibility of allowing negativeX will prove to be useful.

Condition E1: For each x0M , there are constants r > 0, > 0 and Rsuch that for all x, yB (x0 , r ) and t[0, )

d (X t (x) , X t (y)) d (x, y ) (1 + t) .Condition E2 :

d (X s + t (x) , X t (X s (x))) = O (st )

for 0 t s as s 0, locally uniformly in x.Corollary 29 Let X be an arc eld satisfying E1 and E2 on a locally complete metric space M . Then given any point x M , there exists a unique solution x : [0, x ) M with initial condition x (0) = x.

Proof. Follow the proof of Theorem 12.


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20 CHAPTER 1. FLOWS

Corollary 30 Let X be an arc eld on a complete metric space M , which satises E1 and E2. Solutions x : [0, x )

M satisfy

x (s ) (t) = x (s + t)

for s, t 0 and s + t < x . Dening W M R+ by W : = {(x, t )|t[0, x )} and F : W M by F (x, t ) := x (t)

we know F is continuous and (i) M {0} W and F (x, 0) = x for all xM (identity at 0 property )(ii ) F (t, F (s, x )) = F (t + s, x ) (1-parameter local semi-group property )(iii ) For each xM , F (x, ) : [0, x ) M is the maximal solution x to X .If in addition X has linear speed growth, then F has domain W = M R+ ,i.e., F is a forward ow .

Proof. Use Corollary 29 and adapt the proof of Theorem 22.

Theorem 31 Let X be an arc eld on a complete metric space M, which has linear speed growth and satises Conditions E1 and E2, with < 0 uniformly valid for al l of M . Then the forward ow F : M [0, ) M of X has a unique xed point. That is, there exists pM , such that for all t 0, F ( p,t) = p, and if F (x, t 0) = x for some t0 > 0, then x = p. Furthermore, the ow converges tothe xed point exponentially :

d (F (x, t ) , p) = d (F (x, t ) , F ( p,t)) et d (x, p) .Proof. Theorem 16 is valid mutatis mutandis and gives

d (F (a, t ) , F (b, t))

eK A t d (a, b) .

Thus, F t := F (, t ) is a contraction mapping for t > 0 on M , and therefore hasa unique xed point, say pt , by the Contraction Mapping Theorem (Theorem119, Appendix A.1). Note pt is a continuous function of t , since

d ( pt , pt ) = d (F ( pt , t ) , F ( pt , t ))

d (F ( pt , t ) , F ( pt , t )) + d (F ( pt , t ) , F ( pt , t )) eK A t d ( pt , pt ) + d (F ( pt , t ) , F ( pt , t )) d ( pt , pt )

d (F ( pt , t ) , F ( pt , t ))1 eK A t

0 as t tby the continuity of F . The 1-parameter local semigroup property of F gives

pt = F ( pt , t ) = F (F ( pt , t ) , t ) = F ( pt , 2t) =

= F ( pt , nt )

for any positive integer n. Hence, pnt = pt and further pi/j = pi = p1 for allpositive integers i and j . Since t pt is continuous and constant on the positiverationals, pt = p1 for all t > 0.

See Chapter 7 for examples in H (Rn ). Theorem 33 in the next section dealswith the more general question of invariant sets instead of just xed points.


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1.3. INVARIANT SETS 21

1.3 Invariant sets

Denition 32 A set S M is dened to be locally uniformly tangent to

X if d (X t (x) , S ) = o (t)

locally uniformly for xS , denoted S X .S is invariant under the ow F if for any xS we have F t (x)S for all t(x , x ).

The next theorem is a metric space generalization of the Nagumo-BrzisInvariance Theorem (Example 11 shows how this generalizes the Banach spacesetting). The bidirectional case is given, but the result obviously holds also forforward ows mutatis mutandis . Cf. [50] for an exposition on general invariancetheorems.

Theorem 33 Let X satisfy E1 and E2 and assume a closed set S M has S X . Then S is an invariant set of the ow F .

Proof. By choosing x S this theorem is an immediate corollary of thefollowing, slightly stronger fact:

Lemma 34 Let x : (, ) U M be a solution to X which meets Condition E1 with uniform constant on a neighborhood U . Assume S U is a closed set with S X . Then d (x (t) , S ) e|t |d (x, S ) for all t(, ) .

Proof. (Adapted from the proof of Theorem 16, due to David Bleecker.)We check only t > 0. Dene g (t) := et d (x (t) , S ). For h 0, we have

g (t + h) g (t) = e( t + h ) d (x (t + h) , S ) et d (x (t) , S )e( t + h ) [d (x (t + h) , X h (x (t))) + d (X h (x (t)) , X h (y)) + d (X h (y) , S )]

et d (x (t) , S )for any yS, which in turn is

e( t + h ) [d (X h (x (t)) , X h (y)) + o (h)] et d (x (t) , S )et eh d (x (t) , y) (1 + h) et d (x (t) , S ) + o (h)= eh (1 + h) d (x (t) , y) d (x (t) , S ) et + o (h) .

Therefore

g (t + h) g (t) eh

(1 + h) 1 et

d (x (t) , S ) + o (h)since y was arbitrary in S . Thus

g (t + h) g (t)o (h) et d (x (t) , S ) + o (h) = o (h) (g (t) + 1) .


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22 CHAPTER 1. FLOWS

Hence, the upper forward derivative of g (t) is nonpositive; i.e.,

D + g (t) := limh0+

g (t + h) g (t)h 0.

Consequently, g (t) g (0) ord (x (t) , S ) et d (x (0) , S ) = et d (x, S ) .

Theorem 33 will be used to piece together local integral surfaces to getfoliations in 3.4. Also see Example 87.

1.4 Commutativity of owsThe following theorem is valid for both bidirectional and forward ows.

Theorem 35 Let X and Y be arc elds on a complete metric space M which satisfy Conditions E1 and E2. Let F and G be the local ows generated by X and Y , respectively. If

d (Y t X t (x) , X t Y t (x)) = o t2 (1.14)

locally uniformly in x then

F s Gt = Gt F s

that is, the ows commute.

Proof. (1.14) means for any x M there exists a neighborhood U :=B (x, ) and a function with limt0

(t) = 0 such that for all y U we haved (Y t X t (y) , X t Y t (y)) t2 (t). By shrinking U if necessary, both arc elds willsatisfy E1 and E2 uniformly on U for some constants > 1 and , and alsothe speeds of X and Y are uniformly bounded by > 0. For the time beingwe assume s and t are sufficiently small so all the compositions of X and Y appearing below remain in U, i.e., |s| , |t| < / (2). In the last paragraph of theproof, continuation will eliminate this restriction on s and t.

The calculations can become a little convoluted, but using Figure **add it!**you might nd it easier to construct your own proof than to read this one.

Let us rst check the theorem in the case s = t. This next estimate is thelinchpin of the proof.

Lemma: d (Y r X r )i (x) , (X r Y r )i (x) r (r ) (1 + r)2i (1.15)


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1.4. COMMUTATIVITY OF FLOWS 23

Let us verify this estimate. Denote x j := ( Y r X r )j (x)

d (Y r X r )i (x) , (X r Y r )i (x)

i1k=0

d (X r Y r )k (Y r X r )ik (x) , (X r Y r )k+1 (Y r X r )i

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Continuous Dynamics On Metric Spaces by Craig Calcaterra