The Fredholm Alternative for Functional Differential Equation Of

Home , Fredholm alternative

The Fredholm Alternative for

Functional Dierential Equations of Mixed Typ e

John MalletParet

Division of Applied Mathematics

Brown University

Providence RI August

Abstract

We prove a Fredholm alternative theorem for a class of asymptotically hyp erb olic linear dierential

dierence equations of mixed typ e We also establish the co cycle prop erty and the sp ectral ow

prop erty for such equations providing an eective means of calculating the Fredholm index Such

systems can arise from equations which describ e traveling waves in a spatial lattice

Intro duction

Our interest in this pap er is the linear functional dierential equation of mixed typ e

x A x r h

j j

Generally nonlinear nonlo cal dierential equations arise in the study of traveling waves in domains

with nonlo cal interactions such as on a spatial lattice see for example and equations

such as arise as their linearizations ab out particular solutions The p erturbation stability and

bifurcation theory of such solutions via the LyapunovSchmidt pro cedure relies up on a Fredholm

alternative for equation to which the present pap er is devoted

With the exception of early work of Rustichini not much is known ab out general prop erties

of such linear equations In Sacker and Sell consider a general class of innite dimensional systems

and study prop erties related to exp onential dichotomies Although our results are somewhat in this

spirit equation diers in that it is not an evolutionary system that is initial value problems are

not wellp osed

In a companion pap er our results are used to study the global structure of the set of traveling

wave solutions on a spatial lattice We employ there a Melnikov metho d much in the spirit of

earlier work involving global continuation techniques See the survey articles for a broader

outline of the problems and for additional references

We make as a standing assumption for that the d d complex matrix co ecients A

J C are measurable and uniformly b ounded on some usually innite interval J and that the

d p

inhomogeneity h J C is lo cally integrable We shall later imp ose other conditions in particular L

b ounds on h The quantities r the socalled shifts can b e of either sign As a technical convenience

we assume that

r r r j k N N

j k

that is the shifts are distinct with r and hence with r for j N We take these as

standing hyp otheses throughout this pap er In fact the conditions are not restrictions on the

form of the equation but merely a matter of notation since any co ecient A is p ermitted to

vanish identically on J Denoting

r min fr g r max fr g

min j max j

j N j N

we observe that r r and r r

min max min max

We may write equation as

x L x h

where L for almost every J denotes the linear functional

L A r C r r C

j j min max

d d

from C r r C into C When the function h is absent then we have the homogeneous

min max

system

x L x

A sp ecial case of o ccurs with a constant co ecient op erator

A r L

j j

where each A is a xed matrix Here equation takes the form

x L x h A x r h

j j

or simply

x L x

in the homogeneous case

Our notation parallels very closely the standard notation of delay dierential equations for exam

ple as set forth in the b o ok of Hale and Verduyn Lunel where we denote x x for

r r Observe that for almost every the norm of the linear functional in is

min max

kL k jA j

and also that for any b ounded function x L L R C on the line we have the p ointwise

estimate

jL x j kL kkxk

Asso ciated to equation we have the linear op erator dened by

x x A x r

L j j

for appropriately dierentiable x R C when we take J R As we are assuming a uniform

p p d p

b ound for the co ecients A it follows that h x b elongs to L L R C whenever x W

j L

p d

W R C that is

p p

W L

is a b ounded op erator for p Thus equation on the real line R takes the form x h

when b oth the inhomogeneous term h and the solution x b elong to the appropriate function spaces

Understanding the op erator is the main concern of this pap er In particular we shall prove

as a main result a Fredholm alternative for in the case that the asso ciated homogeneous equation

is asymptotically hyp erb olic We also establish related results such as the co cycle prop erty and the

sp ectral ow prop erty by which the Fredholm index of can b e eectively calculated We have the

following theorems

Theorem A The Fredholm Alternative Assume the homogeneous equation is asymptot

p p

ical ly hyperbolic Then for each p with p the operator from W to L is a Fredholm

p p

W of is independent of p so we denote K K and similarly operator The kernel K

L L

p p

L of associated to the adjoint L The range R for the kernel of the operator K K

L L L

L L

in L is given by

fh L j R g y h d for al l y K

In particular

p p

ind ind dim K co dim R co dim R dim K

L L L L

L L

where ind denotes the Fredholm index

Final ly when L L is a hyperbolic constant coecient operator we have

ind dim K co dim R

L L

0 0

and so is an isomorphism

We shall discuss and dene several of the concepts o ccurring in the statements of Theorem A and

also the theorems b elow more formally in the next sections In particular the adjoint L of L will b e

dened it is related to the adjoint of the op erator but with a sign change

p p

W L for each p and similarly In the statement of Theorem A we have that K K

L where p q and this implies that the integral in In particular K for K

L L

decay exists In fact we show in a remark following Prop osition that the elements of K and K

L L

exp onentially at

Theorem B The Co cycle Prop erty Assume the homogeneous equation is asymptotical ly

hyperbolic Then the Fredholm index of depends only on the limiting operators L namely the

limits of L as Denoting

ind L L

we have that

L L L L L L

for any triple L L L of hyperbolic constant coecient operators

The ab ove result in particular equation is termed the co cycle prop erty For an ordinary

dierential equation x A x it is the case that

u u

L L dim W L dim W L

where dim W L denotes the numb er of unstable eigenvalues of the related constant co ecient

system and thus the co cycle prop erty holds The same formula holds for delay dierential

equations that is when all the shifts r are nonp ositive Formula thus allows direct

calculation of the Fredholm index ind in these cases

Unfortunately in the general case of arbitrary shifts r R considered here is not valid in

u u

general in particular b ecause dim W L is p ossible even if one of the quantities dim W L

is nite and the other is innite must fail Nevertheless the Fredholm index can b e eectively

calculated using the sp ectral ow formula given as in the following result Roughly this

gives the index as the net numb er of eigenvalues which cross the imaginary axis along a homotopy

b etween the constant co ecient op erators L and L

Theorem C The Sp ectral Flow Prop erty Let L for be a continuously varying

oneparameter family of constant coecient operators and suppose the operators L L are

hyperbolic Suppose further there are only nitely many values

f g

of for which L is not hyperbolic Then

L L crossL

is the net number of eigenvalues of which cross the imaginary axis from left to right as increases

from to

To dene crossL more precisely x any as in in the statement of Theorem C and

j j

let f g denote those eigenvalues of the corresp onding equation with L L on the

imaginary axis Re We list these eigenvalues with rep etitions according to their multiplici ty

as ro ots of the characteristic equation Let M denote the sum of their multiplicitie s For near

with this equation has exactly M eigenvalues counting multiplicity near the

j j j

L R L R

imaginary axis M with Re and M with Re where M M M The net

j j j j

R R

crossing numb er of eigenvalues at is thus M M We dene

j j

R R

M M crossL

j j

This pap er is organized as follows Following Sections and in which notation and prelimi

nary results are develop ed we construct in Section the Greens function for a hyp erb olic constant

co ecient system This result given in Theorem is used in Section to provide fundamental

estimates on solutions of the general asymptotically hyp erb olic system and ultimately to prove

Theorem A in that section Section is devoted to the pro of of Theorem B Following Section which

provides additional asymptotic information ab out solutions of we prove Theorem C in Section

The pro of of this result the sp ectral ow prop erty involves making generic approximations to one

parameter families of systems and in the App endix we prove the technical result Prop osition

that such approximations are p ossible

Acknowledgment This work was partially supp orted by NSF Grant DMS and by ONR

Contract NJ

Preliminaries and Notation

By a solution of on an interval J we mean a continuous function x J C dened on the

larger interval

J f j J and r r g

min max

such that x is absolutely continuous on J and satises for almost every J Thus for such x

we have that x C r r C for every J We shall always take J R unless otherwise

min max

noted

We shall o ccasionally use subscripts to index a sequence of solutions such as x Although this

notation is ambiguous coinciding with the notation of a translate the meaning should always b e

clear from the context In such a case we shall denote the translate by x C r r C

n min max

namely x x

n n

p p d p

We shall write simply L for the space L R C of L vectorvalued functions on the line par

ticularly when the dimension d is clear We always assume that p We denote the space

p p p

W ff L j f is absolutely continuous and f L g

and recall the continuous emb edding

W L p

where in addition

lim f f W p

Generally if X is a Banach space then we shall denote by kxk the norm of an element x X

although we may simply write kxk when the space X is understo o d If also Y is a Banach space then

we let LX Y denote the Banach space of b ounded linear op erators T X Y and we denote the

op erator norm by kT k Denote the kernel and range of T LX Y resp ectively by

LX Y

K T fx X j T x g RT fy Y j y T x for some x X g

Let us recall that T is a Fredholm op erator in case

i the kernel K T X is nite dimensional

ii the range RT Y is closed and

iii RT has nite co dimension in Y

For such an op erator the Fredholm index is dened to b e the integer

ind T dim K T co dim RT

The adjoint equation of is the equation

y L y A r y r

j j j

where

L A r r C r r C

j j j max min

with A r denoting the conjugate transp ose of the matrix A r Quite generally we let

j j j j

P denote the conjugate transp ose of a matrix P We dene the adjoint op erator of to b e

A r y r y y L y y

j j j

that is It is elementary to verify that

Z Z

p q

y x d y x d x W y W

where p q and where the pro duct of the vectors inside the integrals in is the usual

dot pro duct Verifying equation involves in particular the identity

Z Z

y x d y x d

which follows by integration by parts and from the fact that x as provided that

p p

p and similarly for y In the case of the op erator LW L we denote

p p

R L K W R K

L L

for the kernel and range as in the statement of Theorem A and also

p p

R L R K W R K K

L L

for the kernel and range of the asso ciated adjoint op erator

We denote the convolution of two functions f and g by f g given by

f g d R f g

The functions f and g can b e matrix or vectorvalued

We recall the formula for the Fourier transform f L of a function f L and the formula for

the inverse transformg L of any g L namely

Z Z

i i

e g d e f d g f

With the normalization in we have that

Z Z

b b

hf g i f g d hf g i f g d

for the L inner pro duct of two functions and of their transforms

Some care is needed in dening the integrals if f or f is not integrable as these integrals by

themselves are not welldened Generally for f L consider any sequence and let

g e f d

b b

f almost everywhere Then g f in L in particular there is a subsequence n such that g

Of course for many functions it is not necessary to pass to a subsequence If f L we shall say

that the integrals in are integrals in the Fourier sense

We recall that the Fourier transform takes convolutions into pro ducts and so f g f g

for almost every

d a

If for some a R the function f C satises a growth condition f O e as

we use the usual big oh notation which here means that e f is b ounded as

then we may dene the Laplace transform f of f to b e

f s e f d

for complex s satisfying Re s a In fact f is holomorphic for such s The inverse transform is

given by the integral

k i

f e f s ds

k i

for any k a with The integral along the vertical line Re s k is to b e interpreted

as an integral in the Fourier sense Indeed let g L b e given by g e f for and

by g for Then g f k i and one easily derives from the inverse Fourier

transform in

If g is a function which is meromorphic in a region of the complex plane and if C is a p ole of

g in that region then we recall the denition

resg g s ds

jsj

of the residue of g at Here is a suciently small quantity

If f O e as and if for some b a the Laplace transform f is meromorphic in the

closed halfplane Re s b and is holomorphic on the vertical line Re s b then often we may shift

the path of integration in to the line Re s b picking up the appropriate residues to obtain

e e

f s ds e f rese f

where for any C we let e C C denote the function

e s e s C

The sum in is taken over all p oles of f in the strip b Im s a A sucient condition in

order to shift the path of integration and so obtain the formula is that in addition to the ab ove

i f has only nitely many p oles in the strip b Re s a

ii f s uniformly in the strip b Re s k as jIm sj and

iii the function f b i b elongs to L

Finally let us recall the space S of temp ered distributions which is the dual of the space S of

Schwartz functions that is the rapidly decreasing C functions R C We recall in particular

that the Fourier transform T and the inverse transform T of a temp ered distribution T S are given

b b

T T T T

the factors arising here from the normalization in and the inner pro duct formulas For

more details see for example Rudin

Asymptotically Autonomous Systems

Asso ciated to the constant co ecient system with is the characteristic equation given by

det s where

s sI A e

L j

We say that this constant co ecient system or more simply that L is hyp erb olic in case

det i R

We have that det s at some s if and only if x e v is a solution of for some

vector v Let us observe that we have the asymptotic formula

s sI O jIm sj

uniformly in each vertical strip jRe sj K and so there are only nitely many such ro ots in any

such strip

Asso ciated to the constant co ecient equation is the closed op erator A dened on a dense

domain D C r r C and given by

min max

A D f C r r C j L g

min max

The sp ectrum A C of A is known to consist only of p oint sp ectrum and to coincide with

For each A the generalized eigenspace the solutions of the equation det

E C r r C of A corresp onding to is nite dimensional and consists precisely of those

min max

functions of the form

e p r r

min max

where p is any p olynomial with the prop erty that x e p satises equation for R

We shall refer to such nontrivial solutions x as an eigensolutions corresp onding to the eigenvalue

more generally any nite sum of such solutions for eigenvalues F in some nite set F A

will b e called an eigensolution corresp onding to the set F An eigenvalue is called simple if it is a

simple ro ot of the characteristic equation det s For pro ofs of the ab ove facts and further

background see Rustichini

We shall often write the op erator L in as a sum

L L M

of a constant co ecient op erator and a p erturbation term

M B r

j j

which generally will b e small in some sense When L is written as the sum then we have

L L

p p p p

where by M here we mean the op erator M L L or M W L given by

M x B x r x L

j j

for R In this setting A A B for all j We say that the system or simply that

j j j

L is asymptotically autonomous at in case there exists such L as in for which

lim kM k

In this case of course

lim A A j N

j j

and is called the limiting equation at If in addition this limiting equation is hyp erb olic

then we say that is asymptotically hyp erb olic at We analogously dene asymptotically

autonomous and asymptotically hyp erb olic at If is asymptotically autonomous at b oth

then we say that is asymptotically autonomous and similarly with asymptotically

hyp erb olic If is asymptotically autonomous then the limiting equations at need not b e

the same so that we have

L L M L M

N N

X X

L A r M B r

j j j j

j j

with the limits

lim A A lim A A j N

j j j j

and thus

lim kM k lim kM k

In the case of the constant co ecient system with it is easy to verify that we have

s s and hence that

L L

det s det s

L L

or more simply that

det s det s

L L

if the co ecients matrices A are all real In this case we see that s is an eigenvalue of if

and only if s is an eigenvalue of the adjoint system In any case the system is hyp erb olic

if and only if its adjoint is hyp erb olic Therefore a nonautonomous system is asymptotically

hyp erb olic if and only if its adjoint is asymptotically hyp erb olic

We close this section with the following result which in the case of a delaydierential equation

has b een termed a folk theorem and was proved by Levinger

Prop osition Let C be an eigenvalue of the constant coecient system Then the di

mension of the generalized eigenspace E of al l eigenfunctions e p where p is a polynomial

s equals the multiplicity of s as a root of the characteristic equation det

We omit the pro of of this result as it is identical to that in save for notation

The Greens Function for a Constant Co ecient System

Consider the constant co ecient system and assume this system is hyp erb olic that is

holds We shall construct the Greens function G R C for as follows First observe that

i O j j as and so the function i b elongs to L Thus we may

L L

0 0

take its inverse Fourier transform set

e i d G

with this integral taken in the Fourier sense The function G so dened b elongs to L and with the

following theorem we see that it is the Greens function for

with L given by Assume the hyperbolicity condition Theorem Consider the operator

p p

for the characteristic function Then is an isomorphism from W onto L for

p with inverse given by convolution

G h d h G h

with the function which moreover enjoys the estimate

a j j

jG j K e R

h for some K and a In particular for each h L there exists a unique solution x

W to the inhomogeneous equation

Pro of Interpreting G as a temp ered distribution we consider the temp ered distribution given by

G A G r

j j

For convenience we write as if were a function even though it actually may not b e one The

Fourier transform of is therefore the temp ered distribution

i r

b b b

i I A e i G I G

j L

the identity matrix Thus I where denotes the delta function distribution and therefore

I that is

G A G r I

j j

in the sense of temp ered distributions It follows immediately from that as a function G is

absolutely continuous for all and satises

G A G r almost everywhere

j j

At the function G p ossess left and righthand limits G and G and there is a jump

discontinuity

G G I

We next show that the function G decays exp onentially at b oth To see this rst write

s s I Rs

One sees that Rs is holomorphic and satises the estimate jRsj O jIm sj uniformly in some

strip jRe sj a ab out the real axis as jIm sj We have now by substituting into

that

G E e Ri d

where E is the inverse Fourier transform of the function i I namely

e I

If then we may shift the path of integration of the integral in to the line Re s a to

obtain

G E e Ra i d

The absolute convergence of the integral in ensures that G decays exp onentially at A

similar argument works for and we obtain for some K

p p p

Let us now solve the inhomogeneous problem in the space W for h L Given h L

dene x by the convolution x G h Then by Youngs inequality

p p

kG k 1 khk kxk

L L

We regard the function x as a distribution and note that it is enough to show that equation

holds in the sense of distributions in order to conclude that actually x W and that holds

for the function x almost everywhere Thus we must show that

Z Z Z

h d A x r d x d

j j

for all test functions that is all C functions R C of compact supp ort By Fubinis theorem

and then by and the jump condition we have that

Z Z Z

N N

X X

A G r h d d A x r d

j j j j

j j

Z Z

G d h d

Z Z Z

h d G d h d

Z Z

h d x d

p p

W L is onto x h and so which gives It follows in particular that

L L

0 0

x for some x W is onetoone Supp ose that All that remains is to show that

L L

0 0

Then x satises the homogeneous equation almost everywhere Interpreting x as a temp ered

distribution we have for its Fourier transform x that

i r

b b

x A e i x

b b

i x From the sp ectral condition we conclude that x is the zero distribution hence

and hence that x is the zero function This completes the pro of

The Pro of of Theorem A

In this section following a sequence of intermediate results we prove Theorem A To b egin consider

the op erator with L L M written as the sum of a constant co ecient op erator

and a p erturbation Assume that the constant co ecient system corresp onding to L

is hyp erb olic We do not necessarily assume that L is asymptotically autonomous as in but we

will assume at least initially that kM k is uniformly small for all R By Theorem we may

and hence invert and using we obtain I M

L L L

0 0

L L L

0 0

p p

k p p holds as an op erator from L to W provided that kM

LL L

We wish to obtain a Greens function for such a p erturb ed op erator and in order to do this

we must suitably ma jorize the Neumann series p ointwise in The following technical lemma

will help in this direction

aj j

Lemma Let R be given by e for some and a and let

j j j

R denote the j fold convolution of with itself namely for j

and Then we have that

j a j j

e R

a a a

provided that a

Pro of The Fourier transform of is

and so as long as a we have

a a

a a a

By taking inverse transforms of the as a calculation shows with the ab ove sum converging in L

ab ove the formula is obtained as desired

We now obtain the Greens function for a uniformly small p erturbation of a hyp erb olic constant

co ecient system

Prop osition Suppose the constant coecient operator L in is hyperbolic Then there exist

positive constants K and a such that if L L M as in and if

kM k R

p p

then W L is an isomorphism for p Moreover there exists a function G R

C satisfying the pointwise estimate

aj j

jG j K e R

such that

G h d h

for any h L

we Pro of With G as in the previous section denoting the Greens function corresp onding to

have that

h d h L h M

B G r

j j

Using the b ound we have the p ointwise estimate

a jr j a j j a j j

0 j 0 0

j j jB jK e e K e

where

a jr j N

0 j

sup kM k maxfe g K K

More generally we may write

p j

h d h L h M

where the kernels are dened inductively by

s s ds j

j j

A p ointwise upp er b ound for each is obtained from the b ound namely

j a j j

j j K e

in the notation of Lemma which yields

a j j

j j K e

K a

K a a K a

provided that K a that is provided

sup kM k

a jr j

0 j

K maxfe g

We may therefore set

s ds G s G G

whenever holds and observe that

a j sja js j a j j

0 1 0

K K e ds jG j K e

a j j

K e

for all R and some K where the second inequality in is a consequence of the fact

that a a

One sees that this function G is the Greens function for the p erturb ed equation x h that is

h and let holds for every h L In particular let x

M h x

L L

0 0

denote the partial sum in the Neumann series Then x x in L and x is given by

k k

G h d x

k k

G G G s s ds

k j

Letting y denote the integral expression on the righthand side of we must show that x y

and to this end it is enough to show that x y in L We have the p ointwise b ound

jx y j jR h j R G

k k k

j k

p p

by and hence kx y k kR k 1 khk Finally

k L k L

kR k 1 kG k 1 kk k

k 1

L L

j k

so x y in L as desired

We may use the ab ove result to study op erators L with more general p erturbations M which

need not b e small everywhere We do require that M b e small as j j and in fact we

consider asymptotically hyp erb olic systems We may think of the following result as in some sense

compactifying the real line for solutions of such systems in that uniform decay rates at innity are

of the kernel the maximum of jx j is always achieved provided In particular for elements x K

at in some uniformly b ounded interval

Prop osition Assume that equation is asymptotical ly hyperbolic at Then there exist

p p

positive quantities K K and a such that if x h for some x W and h L then we have

that

aj j aj j aj j

1 p 1

e h d K e kxk K khk jx j K e kxk K

L L L

for

If equation is asymptotical ly hyperbolic that is asymptotical ly hyperbolic at both then

the estimate holds for al l R In addition there exists K such that

1 p

kxk 1p K kxk khk

L L

for any such x and h

Pro of Observe that the second inequality in follows immediately by applying Holders inequality

to the integral in that formula Let us therefore prove the rst inequality in for assuming

that equation is asymptotically hyp erb olic at We write L L M where kM k

We shall use Prop osition in the pro of of our result We remark that the quantity K in the

inequality is not necessarily the same as K in the statement of Prop osition although the

quantity a in will b e the same To x notation let us denote by K and a the constants

app earing in the statement of Prop osition for the op erator L We note that these constants

dep end only on L

Fix such that kM k for all let

and set

L L M M M

We therefore have that

x L x M x h

whenever x h and moreover as k M k for all R we see that the op erator L

satises the conditions of Prop osition Therefore letting G R C denote the Greens

we have for every R that function for

G M x h d x

Z Z

G h d G M x d

where we use the fact that M for all We have then for any that

Z Z

aj j aj j

jx j K e kM kkxk d K e jh j d

Z Z

aj j aj j

e jh j d e d K K K kxk

where K denotes the supremum of kM k for all R One sees directly from this that it is

p ossible to cho ose K so that the desired estimate holds for all

A similar pro of of for holds if equation is asymptotically hyp erb olic at Thus

if is asymptotically hyp erb olic at b oth certainly holds for all R

Now let us prove assuming asymptotic hyp erb olicity at b oth We rst take the L norm

of x in and apply Youngs inequality to the convolution integral in to yield

p 1 p

KK khk KK kxk kxk

L L L

p aj j

where K and K are the L and the L norms of the function e The dierential equation

and the b oundedness of the co ecients A for R now imply that x L with

p p p

kx k K kxk khk

L L L

for some K dep ending only on L Combining and gives the desired result

for some p Remark Supp ose that equation is asymptotically hyp erb olic and that x K

that is x W b elongs to the kernel of Then by with h the zero function we have that

p p

aj j

x O e as and so x K for every p Thus the kernel K is indep endent of p as

L L

claimed in the statement of Theorem A and we may denote this space simply by K

Using Prop osition we have the following lemma

Lemma Assume that equation is asymptotical ly hyperbolic Suppose for some p there are

p p p

sequences x W and h L with x h such that the sequence x bounded in W and

n n L n n n

p p

x converging in W to some element with h h in L Then there exists a subsequence x

n n

x with necessarily x h

Pro of First let

Z Z

h d h d H H

n n

and observe the convergence H H which is uniform on compact intervals This implies that for

each compact interval the sequence of functions H is equicontinuous

Let us next observe that the sequence x is b ounded in L since it is b ounded in W Up on

writing the dierential equation for x as

x H L x

n n n

we conclude from the uniform p ointwise b oundedness of the righthand side of that the sequence

of functions x H is equicontinuous It follows therefore that the sequence x is equicontinuous on

n n n

each compact interval

We may therefore assume by taking a subsequence if necessary which we continue to denote by

x for convenience that x x uniformly on compact intervals for some continuous function

n n

d p p

x R C Certainly we have that x L L since x is b ounded in b oth L and in L Up on

writing the dierential equation for each x in integrated form

x x L x h d

n n n n

and taking the limit n we obtain an integral equation for x and conclude in a standard fashion

that x is absolutely continuous and satises

x L x h

p p

almost everywhere From we have that x L and hence that x W and so x h

It remains to prove that we have convergence x x in the space W From of Prop osi

tion applied to the dierence x x we have that

p 1

kh h k K kx x k kx x k

n L n L n

As h h in L we see that it is enough to prove that x x in L in order to conclude that

n n

x x in W By the second inequality in

aj j

1 p

jx x j K e kx x k K kh h k

n n L n L

hence for any

p 1

K kh h k sup jx x j K e kx x k

n L n n L

j j

o n K e K kx k

where K is such that kx k K for all n and we employ the usual little oh notation As we

n L

already know that x x uniformly on we conclude from that

1 1

lim sup kx x k K e K kx k

n L L

as desired As is arbitrary we have that kx x k

n L

Corollary Assume that equation is asymptotical ly hyperbolic Then the kernel K of the

operator is nite dimensional

Pro of We shall prove that the unit ball

g B fx W j x K and kxk

in the subspace K W is compact As the unit ball in any Banach space X is compact if and

only if dim X see for example Theorem of Rudin we conclude that dim K

The choice of p in our argument do es not matter as long as p since the space K do es not

dep end on p

Take any sequence x K with kx k 1p Then by Lemma with h there exists

n L n n

x converging in W to some x with x Thus x B hence B is a subsequence x

compact as desired

Corollary Assume that equation is asymptotical ly hyperbolic Then for p the

range R L of is closed

L with Pro of Let p b e xed as in the statement of the corollary and take a sequence h R

h h in L We must show that h R

p p

Let C W b e a closed subspace complement of the kernel K that is with W K C

L L

Such a subspace exists since K is nite dimensional by Corollary Certainly maps C onto R

L L

and so there exists a sequence x C with x h

n L n n

First assume that the sequence x is b ounded in W Then with Lemma we obtain some

as desired x W for which x h This proves that h R

is unb ounded without loss we may assume by taking a subsequence Now assume that kx k

and q h kx k that kx k 1p We shall obtain a contradiction Let y x kx k

1p 1p n n n n n n n

W W

Thus y q with y C satisfying ky k 1p and with kq k Again with Lemma

L n n n n n L

we have after passing to a subsequence that y y in W with y Thus y K and

n L L

as y C we have also that y C and hence that y K C fg However ky k 1p since

n L

and this is a contradiction ky k

We now prove Theorem A

Pro of of Theorem A Assume the hyp otheses of the theorem namely that equation is asymp

K of is indep endent of p and we have totically hyp erb olic As noted earlier the kernel K

L L

L is closed by Corollary that dim K by Corollary Also R

L denote the subspace Let Q

fh L j Q g y h d for all y K

in the righthand side of Again as noted earlier the adjoint equation is also asymptotically

hyp erb olic hence the subspace K has nite co dimension W is nite dimensional and so Q

co dim Q dim K

in L Therefore in order to establish Theorem A and in particular to prove that prove that is a

p p

Indeed one sees here that follows Q Fredholm op erator it suces to prove that R

L L

from this result and the fact that L L

p p p

with x W we have by Indeed for any h x R Q It is easy to see that R

L L L

that

Z Z Z

y h d y x d y x d

p p p p

so that R Q Assume therefore for some p that R implying that h Q for each y K

L L L L

Let us rst treat the case in which p the case of p is a prop er subspace of Q

b eing considered later Then there exists a nontrivial linear functional on the space L which vanishes

p p

Such a functional is given by an element of the dual but not identically on Q identically on R

L L

q q

space L so there exists some y L such that

y x d for all x W

y h d for some h Q

This will immediately contradict the We shall show that the rst line of implies that y K

second line of in light of the denition of the space Q

The integral in Consider any C function R C of compact supp ort and set x

the rst line of b ecomes after taking complex conjugates

y d

Z Z

A r d y y d

j j

Z Z

d A r y r y d

j j j

The ab ove formula says precisely that the function y satises the adjoint equation in the sense

of distributions It follows immediately that the distribution derivative y is in fact a function with

p p

q q

y L and hence that y W Thus y K Q as desired This now establishes that R

L L

and thereby completes the pro of of Theorem A in the case of nite p

Now consider the case of p We claim that any h L can b e written as h h h where

h R and where h L has compact supp ort If h Q then we have for such a decomp osition

L L

that h R Q hence h Q However as h has compact supp ort we have that h L for

L L L

0 0 0

p p p

every p and so also h Q Now Q R from ab ove and so there exists x W such

L L L

that x h But b oth x h L so x L for the derivative of x from the dierential equation

as desired and hence also h h h R implying that x W Thus h R

L L

To obtain the ab ove decomp osition h h h for h L it is sucient to nd x W such

that equation holds for all suciently large j j say for j j for some Indeed for such

a function x we simply set h x and h h h and note that h has compact supp ort To

obtain such x it is enough to nd x x W such that x satises equation for all

and x satises for all for some Indeed having obtained such x if we set

x x x

where R R is any C function such that for and for one easily

sees that holds for all large j j

Thus given h L we must nd x W such that x satises equation for all

suciently large Obtaining the function x is similar so we omit the details We employ a

construction used in the pro of of Prop osition Sp ecically we let

L L M M M

where L L M with kM k as and where is as in If the quantity

in is suciently large then is an isomorphism of W onto L by Prop osition and

h It is easy to see that x satises equation for all large as required we may set x

The nal sentence in the statement of Theorem A follows immediately from Theorem

The Pro of of Theorem B the Co cycle Prop erty

We now prove Theorem B which includes in particular the co cycle prop erty for the Fredholm

index Our approach here is similar to that of Angenent and van der Vorst

Pro of of Theorem B To prove that the Fredholm index ind dep ends only on the limiting

op erators L consider two asymptotically hyp erb olic equations of the typ e say x L x

for with the form with co ecients A and the same shifts r As we allow the

co ecients to vanish identically requiring the shifts to b e the same is no restriction but only a matter

of notation Assume the limiting op erators at are equal L L denoted L and similarly

assume L L L for the op erators at For dene L L L

Then for each such the equation x L x is asymptotically hyp erb olic hence is a

p p

varies continuously in LW L with Thus the Fredholm Fredholm op erator Moreover

as desired ind is indep endent of and so ind index ind

1 0

L L

Let us now prove the co cycle prop erty Given L L and L as in the statement of the

theorem consider for the system z L z given by

z L z RL Rz

in twice the numb er d of variables where

cos I sin I

d d

B C

x L x

B C

z L z R

B C

y L y

sin I cos I

d d

where is the Heaviside function namely with and where I denotes the d d identity

matrix One easily checks that is asymptotically hyp erb olic for each and that varies

continuously in and so ind is indep endent of One also sees that at the endp oints

of this homotopy the system decouples into separate equations for x and y and the Fredholm

index of can b e explicitly calculated as the sum of the indices of the x and y equations For

any the limiting equation at is given by op erators L and L resp ectively for the x and y

equations At with the limiting op erators are L and L resp ectively while at with

they are L and L resp ectively One therefore has that

ind 0 L L L L

ind 1 L L L L L L

where L L by of Theorem A As ind 0 ind 1 the co cycle prop erty

L L

holds

Asymptotic Behavior of Solutions

In this section we present two additional results concerned with the asymptotic b ehavior of solutions

They are related to our Fredholm alternative results as they concern solutions of asymptotically

hyp erb olic systems

We rst present a result concerning the inhomogeneous constant co ecient system In prepa

ration for this result let us observe that if is any eigenvalue of the corresp onding homogeneous system

and if f is any C valued function which is holomorphic in a neighb orho o d of in C then for

small the function

f x rese e s f s ds

jsj

is an eigensolution of corresp onding to Here e is as in That the function x so dened

satises equation follows easily by direct substitution into using the formulas

Z Z

s s sr

x s f s ds x r s f s ds e s e e

L L

0 0

i i

jsj jsj

That this solution x is an eigensolution is seen by expanding the Taylor and Laurent series

j j

X X

j s

s f s C s e e

L j

j M

ab out s When inserted into the integral in the expansions yield x e p for

some p olynomial p

Prop osition Let x J C be a solution of equation on the interval J for

some R and some h J C Assume for some real numbers a b that

a b

x O e h O e

Then for every we have that

x z O e

where z is an eigensolution corresponding to the set of eigenvalues

F f A j b Re ag

The analogous result for also holds

Pro of Without loss take Up on taking the Laplace transform of equation we obtain

sxs s hs

where x and h are the Laplace transforms of x and h and where

e x r d s x A

j j

We note that the growth assumptions imply that x and h are holomorphic in the halfplanes

Re s a and Re s b resp ectively Also is an entire function and with equation this

implies that x is meromorphic in the halfplane Re s b with only nitely many p oles there Let

us also note that b oth and h are uniformly b ounded on the vertical strip b Re s k for each

k and so we have xs O jsj uniformly as jIm sj on each such strip

For any k a we have the inverse formula

k i

x xs ds e

k i

with the integral taken in the Fourier sense for Assume that is small enough that the

strip b Re s b do es not contain any eigenvalues and also that b k We may

shift the path of integration in to the left to the line Re s b and obtain

x z w where

e e

xs ds e x w z rese

where F is the set As noted ab ove the function z is an eigensolution corresp onding to eigenvalues

in the set F Thus all that remains is to prove that w enjoys the estimate

w O e

b b

Let u e w and v e w Then from the integral formula for w it

follows that u L C and hence that v is integrable on Because z satises the

homogeneous equation we have that w like x satises the inhomogeneous equation and

we conclude that the function v satises a similar equation namely

br b

v b v e A v r e h

j j

The righthand side of is therefore integrable on and from this we conclude that v is

b ounded as This gives the desired result and completes the pro of

In the next result we consider the homogeneous nonautonomous equation and write L as

in We regard M x as an inhomogeneous term h with kM k decaying exp onentially fast

and we employ Prop osition along with a b o otstrap argument to obtain an asymptotic expression

for solutions

Prop osition Let x J C be a solution of equation on the interval J for

some R Assume that is asymptotical ly autonomous at with L written as with

L a constant coecient operator Also assume for some real number a and some positive number

k that

a k

x O e kM k O e

Then either

i there exists b a and such that

x y O e

where y is a nontrivial eigensolution of the limiting equation corresponding to the nonempty

set of eigenvalues with Re b or else

ii for each b R we have that

lim e x

Solutions x which satisfy conclusion ii of the ab ove result and which are not identically zero on

any interval are sometimes known as small solutions or are said to have sup erexp onential

decay

Pro of Dene the quantity

b sup fa a j x O e as g

If b then conclusion ii holds so supp ose that b Cho ose so that

and such that also we have

Re b b b b

for all eigenvalues of the limiting equation Then with h M x the hyp otheses of Prop o

sition are satised with the quantities b replacing a and with k b replacing b in the

statement of that result We conclude from this lemma that

k b

x z O e

where z is an eigensolution of the limiting problem corresp onding to the set of eigenvalues which

satisfy Re k b b

Now observing from that b b k b b holds we may write

z y O e where y is an eigenfunction corresp onding to the set of eigenvalues

satisfying Re b b that is by to the set of eigenvalues satisfying Re b

k b

This now gives the desired formula where again is used to ensure that e

O e To complete the pro of of the prop osition we must show that the eigenfunction y is not

identically zero But if it were zero then we would have x O e which contradicts the

denition of b

The Pro of of Theorem C the Sp ectral Flow Prop erty

One approach to the pro of of Theorem C is to approximate the family L of op erators in the statement

of that result with a generic family We then show that the crossing numb er for the approximation is

unchanged and that the conclusion of Theorem C holds for it

We b egin by setting some notation With C denoting the set of c d matrices let us denote by

dd N

A A A A C

the array of co ecient matrices in and denote paths in this set by

dd N

b etween co ecients A We keep the shifts r xed throughout this section For any

continuous path let

NH f j equation with co ecients

at A is not hyp erb olic g

Thus satises the conditions of Theorem C if and only if NH is a nite set which we

denote as in

dd N dd N

It will also b e useful to intro duce the map S C C dened for each R by

r r

S A A A A I e A e A

N N

The transformation S arises from the change of variables y e x in equation with

With an abuse of notation let S L denote the op erator as in but with the transformed

co ecients S A in place of A One easily sees that

s s

L S L

0 0

and so the transformation S shifts all eigenvalues to the right by an amount

We next identify generic classes of constant co ecient systems and oneparameter families of

systems

Denition We say the constant co ecient equation with or simply the co ecients

themselves satises Prop erty G if there exists at most one R such that i is an eigenvalue

s and if moreover this is a simple ro ot of the characteristic equation det

Remark Bear in mind that the co ecients A in can b e complex matrices so we do not

in general exp ect eigenvalues to o ccur in complex conjugate pairs Indeed by considering complex

systems we broaden the class of homotopies allowed b etween two given systems and this simplies

our pro of of Theorem C

dd N

Denition Let C C b e a smo oth oneparameter family of co ecients for

We say the corresp onding family of equations or simply the co ecients themselves satises

Prop erty G if

i for each equation with co ecients at A satises Prop erty G

ii at equation is hyp erb olic and

iii all eigenvalues of equation on the imaginary axis Re for some

cross the axis transversely with that is Re

Certainly a family of constant co ecient equations satisfying Prop erty G also satises the hy

p otheses of Theorem C In this case we let denote the eigenvalue as in iii ab ove for which

Re with NH as in We shall also denote

j j j

j j

and we observe that

crossL sgnRe

The following result shows that paths satisfying Prop erty G and joining given co ecients A

dd N

C are dense among all continuous paths joining those endp oints

dd N

Prop osition Let C C be such that the corresponding oneparameter family

of dierential equations with satises the hypotheses of Theorem C Assume also that

there exist j k N such that the ratio r r of the shifts is irrational Then given there

j k

dd N

exists C C such that

ii j j for al l and

iii the family of dierential equations with corresponding to satises Property G

The pro of of Prop osition will b e given in the App endix

Remark If is small enough in Prop osition then one has

crossL crossL

for the op erators L and L corresp onding to and Indeed this is simply a consequence of the

fact that as the eigenvalues are ro ots of a holomorphic function in any b ounded region they vary

continuously as a set in the Hausdor top ology

Remark We b elieve that Prop osition is true even without the technical condition on the ratio

r r In any case this condition p oses no barrier to the pro of of Theorem C as it is really only a

j k

matter of notation Even if this ratio condition is not fullled for the family L in the statement of that

theorem one may trivially intro duce an additional shift r with a co ecient A which is

N N

identically zero so that r r is irrational for some j That is we allow our generic homotopy in

j N

Prop osition to pass through the larger class of systems with the extra shift r

With the following two results we see that without loss we may assume that eigenvalues cross the

imaginary axis by means of a rigid shift of the sp ectrum with the op erator S Here and in what

follows D denotes the derivative of a function with resp ect to its k argument

Lemma Let f s and f s for s C R be two d d matrixvalued functions which

are holomorphic in s in a neighborhood of C and which also are C in in a neighborhood of

R Denote g s det f s and g s det f s Assume that

f s f s

identical ly in a neighborhood of s and that

g D g s j

Denote by s and s the unique solutions of g s and g s near s for

near Assume that

Re Re

that is the two roots and cross the imaginary axis transversely with the same horizontal

speed and direction For set

f s f s f s g s det f s

and let s denote the unique solution of g s near s for near and al l

Then for near f g we have that

Re Re if and only if

Pro of Noting that f s and g s are indep endent of by the identity one has that

identically in Thus all that needs to b e proved is that

Re D j

for all and in fact we show the quantity in equals identically in

By implicit dierentiation we have

D j D g j

0 0

where is the derivative of g in and do es not dep end on To calculate the derivative D g

in consider the determinant function D C C given by D P det P Then by

D g j D f D f j

0 0

D f D f j D f j

0 0

where D P Q denotes the derivative of the function D at P in the direction Q One sees now that

the quantity in the second line of is ane in b ecause the argument f of D is

is ane in and so by indep endent of Thus by we have that Re D j

ReD j Re Re

as desired

dd N

Prop osition Let C C be a oneparameter family of coecients for

dd N

satisfying Property G Then there exists another such C C joining the same

e e

endpoints and also satisfying Property G such that NH NH Moreover

e e

at each NH we have that in the notation above with corresponding to the

j j j j

e e

family in the obvious way Thus the eigenvalues of the the two families and cross the imaginary

axis at the same values of and moving in the same direction left or right

In addition the family has the form

e e

j j

for in a neighborhood of each where S is as in That is for the family the eigenvalues

cross the imaginary axis by a rigid shift of the entire spectrum to the left or to the right

for near

j j

e e

Remark Clearly crossL crossL for the op erators L and L asso ciated to and

Pro of Fix small enough that are disjoint intervals for NH

j j j

dd N

and let C b e such that

S j j

j j

with arbitrary for other values of and with as in Letting L and L denote the

op erators asso ciated to and resp ectively let L L L and consider

s s s

for One sees that f s s and f s s satisfy the hyp otheses of Lemma

near s C R In particular is given by a rigid

j j j j j j j j

translation by and so Re Re verifying

j j j

j j

Thus by decreasing if necessary we may assume that the only imaginary ro ot of det s

near s for j j and is s itself at We may in fact assume

j j j j j j

that for j j and there are no other ro ots anywhere on the imaginary axis

To construct let b e C and satisfy

j j

e b e

and set One now easily checks that satises all the required

conditions

Prop osition Suppose that s i with R is a simple eigenvalue of equation and

suppose there are no other eigenvalues with Re Then for R with j j suciently smal l

we have that

S L S L sgn

Remark The ab ove prop osition veries the crossing formula of Theorem C in the case of the

family L S L

Pro of With L as in make the change of variables u W x in with the

j j

weight function W e This leads to a nonautonomous equation u L u which is

asymptotically hyp erb olic with limiting op erators L S L Similarly up on making the change of

variables v W y in the adjoint equation

y L y

one checks that the resulting equation is v L v namely the adjoint of the equation for u In

fact one has that

W W W W

L L L L

where here W denotes the op erator of multiplication by that function

Assume for deniteness that the case of b eing handled similarly We take small

enough that s i is the only eigenvalue of equation in the strip jRe sj Supp ose that

u K is a nonzero element of the kernel of Then u is a b ounded function of R such that

L L

x W u satises for R But then x is also b ounded on R hence x e p

for some nonzero vector p C It follows that u is unb ounded a contradiction and so K fg

if and only if y W v is a We have that v K Now consider the adjoint kernel K

L L

solution of the adjoint equation satisfying the growth condition y O W as

An application of Prop osition implies that y is b ounded on R and hence is the unique up

to constant multiple solution y e q of One sees that v W y is indeed

b ounded and concludes that K is the onedimensional span of v Thus by Theorem A

S L S L ind dim K dim K

L L L

which proves

dd N

Pro of of Theorem C Let C C denote the co ecients corresp onding to the

family L in the statement of Theorem C Without loss by Prop osition and the rst remark

following that result we may assume that satises Prop erty G Let b e as in the statement of

Prop osition Then for any suciently small we have denoting by L the op erator

corresp onding to and by L L L that

j j +1

e e e e

L L L L L L L L

j j

e e

L L

by the co cycle prop erty of Theorem B For each in the interval for j J

j j

and also in the intervals and equation is hyp erb olic and one concludes

j j +1

e e e e

that L L with L L and L L Thus by and then

j j j j

e e e e

of Prop osition we have that L L S L S L and so

j j

J J

X X

j j

e e

L L S L S L sgn

j j

This with and establishes as desired

App endix the Pro of of Prop osition

Here we prove the generic approximation result Prop osition With d xed dene sets G C

dd dd

for k d and H C C by

G fP C j rankP k g

dd dd

H fP Q C C j rankP d

Q is invertible and rankPQ P d g

We have the following results for such sets

dd dd dd

Prop osition X The sets G and H are analytic submanifolds of C and C C respec

tively of complex dimension

dim G d d k dim H d X

C C

The real dimensions of the manifolds in X are obtained by doubling their complex dimensions

Remark If P and Q are d d matrices with rankP d and with Q invertible then one easily

sees that rankPQ P is either d or d Furthermore the rank of PQ P equals d if

and only if there exists a vector z C such that P z and PQ P z that is the range of

P has nontrivial intersection with the the kernel of PQ Using the fact that the range of P is the

orthogonal complement of the kernel of P we have that rankPQ P d if and only if w v

where v w C are nonzero vectors satisfying PQ v and P w

Pro of In this pro of dimension refers to complex dimension Fix P G Then there exist invertible

matrices E and E such that

X P E J E J

Here J is given in blo ck form with the subscripts k and d k denoting the size of the square identity

and zero matrices

Consider any other d d matrix P without loss written as

I A B

P E JE J

C D

in the same blo ck form as J Then rankP rankJ Assume that P is suciently near P that

the matrix I A is invertible Then a vector

z C

in blo ck form b elongs to the kernel of J if and only if I Ax B y and C x D y that is

if and only if

x I A B y D C I A B y

k k

b oth hold In particular rankJ k if and only if this is the case for every y C namely

D C I A B X

Denoting the lefthand side of X by A B C D we regard as an analytic function mapping

k k k dk dk k dk dk dk dk

a neighb orho o d of in C C C C into C The

dk dk

derivative of with resp ect to D at is an isomorphism in fact the identity on C

Thus by the implicit function theorem the set of matrices near J with rank exactly k is lo cally

an analytic manifold of co dimension d k in C that is of dimension d d k The same

holds for the rank k matrices near P and hence for G and we have the rst formula in X

Let us note also from the ab ove argument that in a neighb orho o d of any P G it is p ossible to

cho ose a basis for the kernel of each nearby P G so that the elements of the basis vary smo othly

in fact analytically with P

To prove the claims ab out H x P Q H and consider nearby pairs P Q G C

For such pairs the matrix Q is invertible and so rankPQ d Also rankP d Let

d d

v v P Q C and w w P C b e analytically varying choices of a nonzero vector in the

kernels of the rank d matrices PQ and P namely PQ v and P w From the remark

following the statement of Prop osition X we have that P Q H if and only if w v Now

dene an analytic function of P Q near P Q by setting

P Q w v v v P Q w w P

To establish the claims ab out H it is enough to show that the derivative of the scalarvalued function

at P Q is nonzero For then by the implicit function theorem the set H which is lo cally the

inverse image is a submanifold of G C of co dimension that is H is a manifold of

dimension dim G dim C d as required

We shall in fact prove that

D P Q X

namely that the derivative of with resp ect to the second argument Q is nonzero Keeping P

P E J E xed as in X with k d vary Q E WE where W is near W E Q E

The matrices E and E are kept xed here Letting e col so that J e we have

v E W e and w E e for the elements v v P Q and w w P in kernels of PQ and P

resp ectively Thus

P Q w v e W e X

by a short calculation It is clear from X that the condition X on the derivative holds and in

fact a nonzero derivative can b e achieved by varying the lower right entry in the matrix W With

this the pro of of the prop osition is complete

Lemma X Let f U C C be a square matrixvalued holomorphic function in some

neighborhood U of a point U Suppose that

rankf d f is invertible rankf f f d X

al l hold Then the function g s det f s has a simple root at s

Pro of Clearly g so we must prove that g By replacing f s with E f sE where

E and E are xed nonsingular matrices we may assume without loss that

A B P Q I

f f X f

C D R S

where all three matrices are written in the same blo ck form From the third condition in X we

have that the matrix P in X is invertible Up on dierentiating the determinant of f s row by

row we have from X that the only contribution to g comes from dierentiating the last row of

f s thereby giving

g det D

C D

Supp ose that D Then from the blo ck forms of f and its inverse and from the invertibility of

P we have that CP DR hence CP hence C But this with D implies that f

is not invertible contradicting X Thus g D as desired

With the shifts r xed and with A as in consider the maps

dd N dd dd N dd dd

P Q C R C P Q C R C C

dd N dd dd

R C T C C

given by

N N

X X

i r i r

j j

P A i I A e QA I r A e

j j j

j j

P QA P A QA

RA P A P A

where T is the set

T f R j g

With the ab ove maps and with the aid of Lemma X we have the following characterization of a

generic set of co ecients

dd N

Corollary X Suppose that A C satises al l of the conditions

P A G k d R

P QA G G k d R

d k

P QA H R

RA G G T

d d

for al l ranges of k and indicated Then the constant coecient equation with

satises Property G

Pro of We note that

P A i QA i i

and so the rst line of X implies that rank d whenever for all eigenvalues i with

is invertible for such The third zero real part The second line of X further implies that

d for such and thus s is a simple ro ot of line implies that rank

L L

0 0

s by Lemma X Finally the last line of X implies that the characteristic equation det

there is at most one value R for which det i Thus all conditions in the denition of

Prop erty G are fullled

Let us recall the SardSmale Theorem casting a sp ecial case of it in a form which will b e useful to

us In this discussion including Theorem X by dimension we mean real dimension Recall rst that

a smo oth map F M M b etween manifolds is said to b e transverse to a submanifold M M

on a set S M if

rangeD F x T M T M whenever x S and F x M

F x F x

where T M denotes the tangent space of M at a p oint p In particular if dim M dim M dim M

then transversality means simply that F x M for all x S

Theorem X SardSmale Theorem Suppose that

a c

F R R M

b c

is a smooth map where R is open and M R is a smooth mdimensional submanifold By

smooth here we mean C where r also satises r b m c Assume the map F is transverse

a a

to M on R Fix two points R and let

r a

X f C R j g

the set of smooth paths joining to Then there is a residual set Y X of such paths such that

for each Y the composed map R dened by

is transverse to M on the set In particular if b c m then for such the range

of on this set is disjoint from M that is

Pro of Consider the map X R given by

Of course the map in the statement of the theorem is simply with the rst co ordinate xed

One sees immediately that is C and that for any X we have that

rangeD rangeD F j

It follows that is transverse to M on the set X From the SardSmale Theorem as

given in we have that for a residual set of the map is transverse to M on as

claimed

We wish to apply the ab ove version of the SardSmale Theorem to the four maps o ccurring in X

of Corollary X In order to do that we must establish transversality of each map to the appropriate

manifold In fact we show that if there is a rich enough collection of shifts r then each of these maps

has a surjective derivative and so is transverse to any manifold in the target space In our application

dd N dd N

of the SardSmale Theorem we identify C R and hence C R

dd N

Lemma X For the maps F P and F P Q in X at each point A C R in

their domain the derivative D F A with respect to the rst argument A is surjective

If r r is irrational for some j k N then for the map R in X the derivative

j k

dd N

D RA is surjective at each A C T

Pro of From the formula X one sees directly that the derivative of P with resp ect to A C

is I the negative of the identity on C recall r Similarly the derivative of P Q with

resp ect to A A C is given by the matrix

i r

I I e

d d

i r

I r e

d d

which is an isomorphism on C

Now x T Then at least one of the quantities r or r is irrational

j k

Supp ose that r is irrational Then the derivative of R with resp ect to A A is given by

j j

i r

1 j

I e I

d d

i r

2 j

I I e

d d

i r i r i r

1 j 2 j 1 2 j

which is an isomorphism since e e that is e

dd N

Corollary X Fix any two points A C and assume for some j k N that r r

j k

is irrational Then there is a residual subset Y X of the space of curves

dd N

X f C C j A g

joining these points such that for any Y and for any al l four conditions X hold

for A

Pro of The version of the SardSmale Theorem given in Theorem X implies that for a residual set

Y of curves all the comp osed maps P P Q and R are transverse

to the manifolds app earing in X In particular Lemma X ensures the required transversality

hyp othesis of the SardSmale Theorem is fullled

By counting real dimensions we see that transversality in all these cases means simply that the

map in question never hits the manifold that is the range of the map is disjoint from the manifold In

particular for the map P in the rst line of X we have in the notation of Theorem X that b

the variable and that c m d k the real co dimension of G by Prop osition X

and so b c m A similar check of the remaining three maps in X reveals that b

and in the second third and fourth lines resp ectively of X and that c m d k

and resp ectively in these cases hence c m Thus b c m in every case

Pro of of Prop osition By Corollary X we may assume without loss that the family

in the statement of Prop osition is such that all four conditions X hold for A for

each Thus each such A satises Prop erty G It is enough to p erturb to a nearby

dd N

e e

C C with the same endp oints such that all eigenvalues of the

corresp onding family of equations with cross the imaginary axis transversely with

There exist such that all eigenvalues of equation satisfying jRe j for any

A are simple and by hyp erb olicity at the endp oints that moreover there are no such

eigenvalues for Lo cally such eigenvalues are parameterized as smo oth

functions of I on some maximal op en interval I for which jj There

are at most a countable numb er of such parameterizations denoted say by f g for I

j j

By Sards Theorem almost every is a regular value of all the functions Re Fix

any such Let R b e any smo oth function satisfying and for all

and with j j for all Then recalling the shift op erator

one easily checks that the family S satises all the required prop erties iiii in the

statement of the prop osition In particular the eigenvalues for the p erturb ed family cross

the imaginary axis transversely

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