A New Pro of of the Stable Manifold Theorem

y

Richard McGehee Evelyn Sander

Preprint Octob er

Abstract

We give a new pro of of the stable manifold theorem for hyp erb olic

xed p oints of smo oth maps This pro of shows that the lo cal stable

and unstable manifolds are pro jections of a relation obtained as a limit

of the graphs of the iterates of the map The same pro of generalizes

to the setting of stable and unstable manifolds for smo oth relations

Intro duction

The stable manifold theorem states that for a smo oth map near a hyp erb olic

xed p oint the stable manifold p oints whose forward orbit converges to the

xed p oint and the unstable manifold p oints with backward orbit converg

ing to the xed p oint are b oth smo oth manifolds This pap er presents a

new pro of of the stable manifold theorem The theorem is proved in the con

text of hyp erb olic xed p oints of smo oth relations a generalization

which includes as sp ecial cases hyp erb olic xed p oints of b oth invertible

and noninvertible maps However this is not merely a generalization

of the standard theorem The new approach restores to the noninvertible

case the symmetry b etween the stable and unstable manifolds as is seen in

the dieomorphism case In addition it provides a new geometric wayof

lo oking at the lo cal stable and unstable manifolds of a map namely they

are b oth pro jections of an ob ject one can think of as the innite iterate

of the graph of the map

mcgeheegeomumnedu

y

sandergeomumnedu Both authors at The Geometry Center South Second

Street Minneap olis Minnesota and The Scho ol of Mathematics University of Min

nesota Minneap olis MN

The key to this new pro of is that rather than lo oking at stable and

unstable manifolds as subsets of the state space we view them as pro jections

of a smo oth manifold in higher dimensions arising from the graph of the

original map More preciselynearahyp erb olic xed p oint the graph of

a map and the graphs of its iterates can b e expressed in an appropriate

co ordinate system as graphs of smo oth contractions The limit of these

contractions exists and is smo oth The graph of this limit pro jects to the

stable and unstable manifolds

n

The derivativeofasmoothmapon R atahyp erb olic xed p ointhas

no eigenvalues on the unit circle Thus lo cally in co ordinates given by the

stable and unstable directions X and Y a map can b e expressed as follows

Ax g x y x

By g x y y

where x and y are vectors in X and Y A and B matrices with jAj

jB j and g and g are Lipschitz with small Lipschitz constant

By the Implicit Function Theorem we can lo cally c hange to a skewed

co ordinate system such that in these new co ordinates wehave a lo cal con

traction Namelywe can write

Ax g x y x

y B y g x y

g andg are again Lipschitz with small Lipschitz constant

The pro of presented here capitalizes on the fact that the map and all

its iterates are lo cal contractions when written in this skewed co ordinate

system Before presenting the pro of we illustrate the ideas with some simple

examples

Some Simple Examples

Consider the graph of the following linear dieomorphism on R with

hyp erb olic xed p oint

x a x

for a b

y b y

Since the xaxis and the y axis are resp ectively the onedimensional sta

ble and unstable directions wecho ose them to b e the directions X and Y

resp ectively in the skewed co ordinates Call the new function resulting from

writing f in skewed co ordinates It is written as follows

x x a

y y

b

th

The k iterate of the original map is

k

x a x

k

k

y b y

k

th

Writing the k iterate in skewed co ordinates gives the following function

k

Note that is found bylookingatf and not by iterating

k k

k

a x x

k

y y

k

k

b

Consider the limit of the it exists and is equal to the map whichis

k

identically zero explicitlylim is the following map in skewed co or

k  k

dinates on R

x x

y y

Notice that this limit map in the skewed co ordinate system do es not

corresp ond to a function in the original co ordinates However we can gain

information ab out the stable and unstable manifolds from its graph Namely

the pro jection of the graph to the xy plane is the xaxis the stable mani

fold The pro jection of the graph to the x y planeisy axis the unstable

manifold

The trick in Example still works if the linear map is noninvertible

ie if a The map b ecomes

x

for b

by y

which can still b e expressed in the same skewed co ordinates as b efore

x

y y

b

th

The limit of the thek iterate written in skewed co ordinates is the

k

same as b efore Indeed the stable and unstable manifolds are once again

the xaxis and y axis resp ectively

If we allow the stretching term b in Example to increase without

b ound the graph of the map converges to fu au v u v R g This

is no longer the graph of a function but is only a relation

Denition Relation Arelationonaspace Z is a subset of Z Z

Viewing this in terms of iteration an iterate of z under relation F is a point

z such that z z F Notice that iterates of a point arenotnecessarily

unique nor do iterates necessarily exist

The relation in this example is a twodimensional plane which is a subset



of R with second co ordinate always equal to A p ointx y R has no

iterates unless y A pointx has as iterates every p oint of the form

ax y y RThus the origin is still a xed p oint under iteration Since

th k

points on the xaxis have k iterates of the form a x whichconverge

to the origin the xaxis is in and in fact equal to the stable manifold

Likewise every p ointonthe y axis is an iterate of the origin Thus the

y axis is contained in and in fact equal to the unstable manifold

We can also use the technique in Examples and to see this although

there is no longer a map limit of b increasing without b ound corresp onds

to b ie Thus although our example is no longer a map it is

b

the graph of a function in skewed co ordinates

ax x

y

In this case as in Examples and the limit of the iterates as expressed

in skewed co ordinates exists and is equal to the zero function Again the

pro jections of the graph of this zero function are the stable and unstable

manifolds

Here is a contrived quadratic example to illustrate the same idea in

a nonlinear case Note that the map f on R has a hyp erb olic xed p oint

x ax

for a b

y by cx

Since the axes are again the stable and unstable directions wecho ose the

axes for the skewed co ordinate directions as b efore The map represented

in the skewed co ordinate system gives the following function

ax x

y y cx

b

Figure shows the graph of with domain By the

fact that isacontraction this gure is the same as the graph of f with

b oth domain and range restricted to Since the graph of

Figure Pro jections of the graph of resulting from the map in Example

Domain and range are a b c Top

left is the xy plane top rightthex y plane b ottom left the xy plane

b ottom rightthe x y plane



a map from R to R is in R the gure consists of pro jections of the graph

to co ordinate planes The pro jections have the following relationship to the

maps f and f maps the region in the xy plane to the region in the x y

plane maps the region in the xy plane to the region in the x y plane

th k

The k iterate f is

k

a

a x x

k

k

where

k

x b y c y

b

k

Represented in skewed co ordinates it gives the following function

k

k

a

x a x

k

k

where

y c x y

b

k

k

b

Figure shows the graph of for the same domain and constants as





in Figure Again f maps the region in the xy plane to the region in

the x y plane maps the region in the xy plane to the region in the



x y plane

The limit lim exists It is given by

k  k

x

c

x

y

a

b

As in the previous examples the pro jections of the limit map to the xy

and x y planes are resp ectively the lo cal stable and unstable manifolds for

f

Since the convergence to the limit function is exp onentially fast the

graph of in Figure is visually indistinguishable from the graph of



lim This is why three of the pro jections app ear to b e curves How

k  k

ever the graphs of b oth and the limit function are twodimensional





surfaces in R To emphasize this p oint Figure shows pro jections of the



same surface after it has b een rotated in R

Wenowshow that the result from the ab ove examples generalizes to

a certain class of relations In Section wegive basic denitions for the

dynamics of relations and state the stable manifold theorem in this general

setting In Section we outline the pro of of the stable manifold theorem

Finally in Section wegive the full details of the pro of outlined in the

previous section

Figure Same pro jections as gure this time of the skewed function



of the twentieth iterate of f This is very close to the limit case Although

three of the pro jections lo ok like curves they are actually pro jections of a

surface See Figure

Figure The graph of shown in Figure after it has b een rotated slightly





in R Same pro jections as b efore This gure illustrates that although three

of the four pro jections in Figure app ear to b e curves the graph is actually



a surface in R

Basic Denitions

In the previous section relations on Z were dened as subsets of Z Z and

were viewed in terms of iteration Here are some denitions in this context

f

We denote z having an iterate z under relation f by z z

Denition Fixed Point Given a relation f on set Z z Z is a xed

point of f if z z f

Denition Comp osition for Relations Given relations g and h on

set Z h g is the relation given by

fz z z Z z z g and z z hg

Notation If I is an interval of integers and z Z for all k I is a

k

sequence of p oints in Z thenwedenote

z z z if I i j

i i j

fz g z z if I j

k j
j

k I

z z if I i

i i

Denition Orbits for Relations Given relation f on space Z an or

I and bit through z is a sequence fz g such that z z for some i

k i

k I

z z f whenever k k I If I i then fz g is cal ledan

k k  k

innite forward orbit If I i then fz g is cal led an innite backward

k

orbit

Denition Stable and Unstable Manifolds For a relation f on met

s

ric space Z with xedpoint z the stable and unstable manifolds W z and

o o

u

W z are dened by

o

s

ough z W z fz Z there exists an innite forward orbit fz g thr

o k

such that z z as k g

k o

s

W z fz Z there exists an innite backward orbit fz g through z

o k

such that z z as k g

k o

r

Denition C Relations If f is a relation on a smooth manifold Z

r r

then f is C when it is a C embedded submanifold of Z Z

Denition Linear Relations If f isarelation in a vector space Z

then f is a linear relation if it is a linear subspaceof Z Z

Denition Hyp erb olic Linear Relations If f is an ndimensional

linear relation on an ndimensional vector space Z thenf is hyperbolic

s u

when there is a splitting Z E E such that under this splitting f is of

the form

x

B C

by

s u

B C

x E y E

A

ax

y

where a and b are matrices and jaj jbj

Note that the graph of anyhyp erb olic linear map is a hyp erb olic linear

relation See Example for the case of a saddle in R

r r

Hyp erb olic Relations A C relation f on a smooth Denition C

manifold Z has a hyperbolic xedpoint z when T f its tangent plane

o z z 

o o

at z z isahyperbolic linear relation

o o

Note that the graph of a map with hyp erb olic xed p oint z is a relation

o

whichhashyp erb olic xed p oint z

o

Wecannow state the main theorem of the pap er

r

Theorem Stable Manifold Theorem for Relations If f is a C

n s

relation on R andf has hyperbolic xedpoint z thennear z W z

o o o

u r

and W z aregraphs of C functions

o

Outline of the Pro of of the Main Theorem

The following denitions and lemmas outline the pro of of the main theorem

The pro ofs of the lemmas are in the next section

n r

First note that for a relation f on R with a C hyp erb olic xed p oint z

o

f is lo cally the graph of a function f More precisely for any and k r

n s u

there is a neighb orho o d of z such that for some splitting R E E on

o

this neighb orho o d f is the graph of function f which is of the following

form

x ax g x y

f

x y y by g

s u

where x E y E a and b matrices jaj jbj and g and g functions

whichhave all derivatives of order k Lipschitz with Lipschitz constant

Motivated by this lo cal expression of a hyp erb olic relation as the graph

of a function we consider some denitions for relations on Euclidean space

Z which are graphs of functions with certain prop erties for some co ordinate

system We call these functions asso ciated functions and call the co ordi

nates skewed co ordinates represented by X and Y where Z X Y and

X and Y are Euclidean Note that not every relation is the graph of such

an asso ciated function these denitions are sp ecically intended for work

ing with relations with hyp erb olic xed p oints Also notice that the skewed

co ordinate system is not unique in any of the denitions b elow However

once wecho ose a co ordinate system if there is an asso ciated function in the

co ordinate system then it is unique

Notation In all that followsarelationisrepresented by a letter and

an asso ciated function for this relation by the same letter underlined

Denition Lipschitz Relations Arelation f is Lipschitz of order

Lip such that orf Lip when thereisanassociated function f

f

x y x y f x y x y

Lemma Suppose a relation f in Lip has an associated Lipschitz

r

function f describedintheabove denition Then the relation f is C exactly

r

when the associated function f is C

The pro of of the ab ove lemma follows from the implicit function theo

rem It is tacitly assumed in the following lemma which states that the

r

comp osition of two Lipschitz and C relations gives another Lipschitz and

r

C relation

r

Lemma Let and r If g relations in Lip and C on

Z X Y with associated functions in the same skewedcoordinates then

r

g Lip and C as wel l

Given relation f for a relation dene G by Gf f The

following lemma says that for f with a hyp erb olic xed p oint and certain

G is a contraction

et f satisfy the hypotheses of theorem and For suitably Lemma L

smal l neighborhood of the xedpoint assume is Lip with associated func

tion in the same skewedcoordinates as f Note that fg lies in the Banach

space of Lip relations in a xedskewedcoordinate system with the norm

being the sup norm on the associated functions Then G is a contraction in

the sup norm on the associated functions

Since G is a contraction in the space of Lip relations G has a unique

xed p ointwhich is also in the space of Lip relations and anysuch relation

converges to this xed p oint In fact we can cho ose a neighb orho o d such

that f is an appropriate Lipschitz relation in the domain of GThus on this

k

neighb orho o d the xed p oint is equal to lim f Call this xed p oint

k 

relation h and its asso ciated function h The ab ove lemma guarantees that

r

h is Lipschitz on In fact h is also C on as is restated b elow

r

Lemma Assume f C satisfying the hypotheses in theorem and for

a neighborhoodof z such that on the xedpoint relation is Lipschitz

o

k r

and equal to lim f Thenh is C on

k 

Denition limit relation Given the relation f on compact metric

space Z

k

f f

n k n

k

where f is the composition of k copies of f

The following lemma states that the relation h dened ab ove is equal to

the limit relation

Lemma f h

The next two lemmas state that limit relation is lo cally the cross

pro duct of the stable and unstable manifolds

Lemma For a relation f satisfying the hypotheses of theorem thereis

s u

a neighborhood of the xedpoint such that if u W z and v W z

o o

then u z and z v arecontainedinf

o o

In fact a stronger statement holds the following lemma states that f

u s

z toevery p ointof W z relates every p ointinW

o o

Lemma For a relation f satisfying the hypotheses of theorem thereis

s u

a neighborhood of the xedpoint such that u W z and v W z

o o

u v f

Pro of of theorem By lemma the stable an unstable manifolds are

s u

pro jections of f Precisely f W z W z By lemma this

o o

set equals fu u z f gfv z v f g By lemma h f

o o

r

In terms of the by lemmas and h has an asso ciated C function h

s u

splitting denote z x y W W fx y hx y x yg

o o o o o

s u

x wz y g Thus b oth W and W are lo cally the graphs fz w h

o o

r

of C functions

Pro ofs of lemmas

r

Pro of of lemma The pro of is an application of the C and Lipschitz

r

implicit function theorems Since it is less common than the C implicit

function theorem we state the Lipschitz version here

Theorem Lipschitz implicit function theorem If X and Y aremet

ric spaces and F X Y X isacontinuous mapping F Lip

then there exists function g Y X g Lip such that

F x y x x g y

Pro ceeding with the pro of of lemma weneedtoshow that if g Lip

r r

and C then there exists a Lip and C function g suchthatx y x y

g exactly when g x y x y Dene a function F Z Z Z

Z Z by

x y x y x y x y gx y F

Since F is Lip and has no unit norm eigenvalues by the implicit function

r

theorem there exists a Lip and C function m Z Z Z such that

x y x y x y F x y x yxy x y x y exactly when m

Thus m m g



Pro of of lemma This pro of is a series of estimates The key to the

estimates is that f and the domain of G are Lipschitz

Assume that f is as in the theorem and wehavepicked a neighb orho o d

and splitting so that equation holds and g g are Lip functions Let

maxjaj jbj Assumewehavechosen a small enough neighb orho o d

that

Note that f Lip

For a relation with asso ciated function letkk denote the sup norm

and let b e the comp onents of the asso ciated function

Wewant to showthatforany relations Lip there is some uniform

constant suchthatkG Gk k k This is equivalentto

showing that sup k where jx y x y j k

xy 

x y x y G and G

If x y x y G and G then there exist

x y x y suchthat

f f

x y x y x y x y

f f

The following inequalities hold

jx j jax g x y a g j

jx j since y

and jx j j x y j

maxjx j jy j k ksince Lip

Similarly

jx j jy j

jy j jx j

jy j maxjx j jy j k k

jy j jy j

If we let maxjx j jy j then from the ab ove equations

wehave

kso k

k k

which is guaranteed by our original assumption Thus for

sup jx y x y j k k

xy 

r r

Pro of of lemma Toshowthath C when f C we rst show that

there is a neighb orho o d of the xed p ointoff such that the limit relation of f

r

restricted to this neighborhood is C To do this we usethebercontraction

theorem to show that the map G is a C contraction when f is C G is

r r

lo cally a C contraction when f C by an induction argument In order to

r

show that h is a C relation on the original neighb orho o d the relationship

between h and the limit relation on a smaller neighb orho o d b ears further

comment To this end weprove that h is equal to the limit relation on

r

the smaller neighb orho o d comp osed with nitely many C subsets of f

r

wing Therefore h is C on the entire original neighborhood We use the follo

denitions and lemmas the central pro of follows their statements and pro ofs

The following is a denition of a derivative relation of a smo oth relation

p

Denition Tangent Relation Given a smooth relation on R the

p

tangent relation T on R is the tangent bund le of

If a relation has an asso ciated function then its tangent relation has an

asso ciated function as describ ed in the following lemma

p

Lemma For a smooth relation on R X Y with associated function

T is the graph of D In other words x y x y T

exactly when x y x y and D x y

Pro of This is due to the fact that a graph of a smo oth function has tangent

bundle equal to the graph of the derivative of the function

Lemma Derivatives and comp osition Assume that and g are

smooth relations with associated functions and x y x y g Then

equal to local ly there exist x and y such that the graph of D g

xy 

graphD In terms of tangent relations local ly graphDg

x y 

xy 

T Tg T g

Pro of of lemma In the pro of of lemma we showed that lo cally there

are a unique x and y which are functions of x y such that

g



x y x y x y

n

x y y For the co ordinate system R g Weknowthaty

D g D g

s u

E E write the derivative matrices in the form Dg

D g D g

Implicit dierentiation gives

Dy D D D D g D g

where all derivatives are evaluated at x y x y x y It is now p ossible

to write the derivativeof g explicitly Comparing this derivative to the

graph Dg shows that they are function asso ciated with graph D

x y 

xy 

equal

r

Lemma Let g and both beLip C relations on the compact

set V assume that in the coordinates V V V thereisacontraction g

associated with g Also assume that for U V and U V g U V

V U V Let bearelation on V Then g g is Lip V and g

r

and C on U U

Pro of of lemma Dene the function F V V V U V

V V as F x y x y x y x y g x y x y gx y

Pro ceed using the implicit function theorem as in the pro of of lemma

The nal lemma is the b er contraction theorem due to Hirsch and Pugh

Its pro of can b e found in

Lemma Fib er contractions Let beamaponaspace X with at

tractive xedpoint pFor each x X let be a map on metric space Y

x

such that x y x y is continuous on X Y For xed

x

and each x let each Lip Then thereisanattracting xedpoint p q

x

for

Finally the following denition makes the notation more convenient

r r r

Denition C and Lip Small Relations Ar elationisC

r

smal l if there is some associated function which is C smal l in other

r

words there is an associated function which is C and al l its derivatives of

r r

order r areLip Arelation is Lip smal l if it is C smal l and the

th

r derivative of the associated function is Lip

r

Using these lemmas the pro of pro ceeds as follows Let f be a C relation

n

on R with hyp erb olic xed p ointatz as in theorem Let b e a

o

j

neighb orho o d of the xed p ointsuch that lim f h as describ ed in

j 

the discussion after the statement of lemma Let and b e as in the

pro of of lemma and let z x y in terms of the splitting

o o o

is C r Since Tf First we show G is a C contraction when f

is not necessarily Lipschitz we cannot just apply lemma on the tangent

bundle However as in the dieomorphism case we can still prove the result

using the b er contraction theorem

n n

Let beaLip relation on R such that L a relation on R

and L x y linear Consider the map TG Tf Tf Weverify

the conditions for the b er contraction theorem for TG TG has attractive

xed p oint h Near x y Df x y is close to Df x y in linear norm

o o o o

Thus we can use estimates similar to those in the pro of of lemma on TG

on a neighborhood of x y On such a neighb orho o d for xed and

o o

varying L TG is Lip in the sup norm By the b er contraction theorem

TG is a contraction in the sup norm on relations ab ove Thus TG is

n

acontraction when T where is a Lip small relation on R

Therefore G is a C contraction on Lip small relations

For the case r pro ceed by induction Assume that for all relations

r
p r

g C on with hyp erb olic xed p ointon R andfor Lip small

r
r

that g g is a C contraction Cho osegC hyp erb olic By our

r
r

assumption T T g T T g is a C contraction when Lip small

r

relations Therefore g g is a C contraction

r

Wehave so far shown that for f C there is an suchthatonaballof

r r

radius of the xed p oint G is a contraction in the C sup norm on Lip

r

small relations The limit relation on this small ball is thus C Wenow

use the smo othness of the limit relation on the small balls to show that the

r

relation h is C on all of

Cho ose r Denote the ball of radius of the xed p ointby B B Let

r

b e the C xed p ointof G restricted to this ball Note that h since

it can b e describ ed as the limit of iteration of the relation f j

B B

For describ ed in the pro of of lemma if jx y x y j then

o o

x y x y j Thus f B B B jf B B B and f

o o

r

B B Thus by lemma f f is Lip and C when restricted to the

B set B

This new relation is also a subset of h since if f f and then

f f Weknowthat hThus f f f h f h

Now iterate this pro cess of comp osing with f and restricting to a neigh

r

borhood eventually wehave a Lip C relation on Since this relation is

contained in h and b oth are asso ciated with functions on the relations

r

must b e equal Thus h is C on

Pro of of lemma Assume that wehave a neighb orho o d and splitting

k

as in f equation Weshow that there is a sequence in f converging

to a limit p ointx y z w exactly when there is a sequence k such that

i

k

i

x w z y lim f

i

First we showthath f from the denition

k

i

f u u u for some u f

k k

i i

By lemma f f f f f maps to h in the sup norm on the

k

x w has a limit and asso ciated functions Thus for all x w andoddk f

k k

x w Thus u x f x w f x w wshows the limit is equal to h

k

that h f

Converselytoshowthatf h supp ose x y z w f and

k

i

u x y z w guaranteed by equation is the sequence in f

k k k k k

i i i i i

k k

i i

x w f x w w Then suchthatju j Dene v x f

k k

i i

j v jj u j j u v j

k k k k

i i i i

The rst term on the rightgoestozeroby construction In addition since f

is in Lip the second term is less than or equal to the rst term Therefore

it go es to zero as well Thus h

Pro of of lemma follows from lemma

Pro of of lemma Assume that wehave a neighb orho o d and splitting

describ ed in equation and that in terms of the splitting the xed p ointis

s u

denoted byx y Assume x y W x y and z w W x y In

o o o o o o

the pro of that follows welookattheforward k iterates of a neighb orho o d

of x y and the backward k iterates of a neighb orho o d of z w For large

near the xed p oint a p ortion of the forward iterates form a Lipschitz k

vertical curve and a p ortion of the backward iterates form a Lipschitz

horizontal curve The two curves are near each other and thus intersect

implying the existence of a p ointnearx y with a k iterate near z w

More preciselywe use this idea to show that for any K there exists kK

k

andapoints t u v f such that distx y z w s t u v and thus

x y z w f

Let b e given Weknow that there exist sequences x y and z w

k k k k

k k

b oth converging to the xed p oint x y x y f and z w zw f

k k k k

For a small let k b e large enough that distance from x y toz w

k k k k

is less than

Nowlookatan ball of y in Y For the p ointx where is in the

k

k

ball wehave the p ointx f The set of p oints form

the graph of a Lipschitz function from Y to X near x y eachpointof

k k

k

which is related to a p ointnearx y by f Similarly there is a graph

of a Lipschitz function from X to Y near z w and a p oint near z w

k k

is related to each of the p oints in this graph But if is small enough

k

hitz graphs must intersect Thus there is a p oints t u v f these Lipsc

within of x y z w We conclude that x y z w f

Conversely assume x y z w f Therefore for any k there

are p oints near x y with k forward iterates Using compactness weshow

that x y has an innite forward orbit Using the fact that f Lip

we show that the forward orbit must converge to the xed p oint and thus

s u

x y W x y Likewise z w W x y

o o o o

Let B x y b e the closed ball of x y and dene the set

k th

S x y B x y hasa k iterate

k

S x y is nonemptyby the assumption on x y It is compact since f

k k

is closed which implies f closed and thus compact Thus S x y is

nonempty since it is the intersection of nonempty nested compact sets It

is equal to fx y g since this is the only p oint it could contain Therefore

th

x y hasa k iterate x y forevery k Thus there exists an innite

k k

forward orbit starting at x y By compactness there exists a limit p oint

us x y z w f for the same z w for all j z w Th

j j

Since x y z w f for all j ifx y and x y are in this

j j k k j j

forward orbit then jy y j jx x j

j k j k

k

Since f is Lip for every k jx x j maxjx x j jy y j

k  k k
k k  k

maxjx x jjx x j Thus jx x j jx x j This Cauchy

k
k k  k k  k k
k

sequence implies that x converges to a unique z Likewise y converges

k o k

to a unique w This means that x y x y z w z w Since

o k k k  k  o o o o

f z w z w f as well The f is closed and x y x y

o o o o k k k  k 

unique xed p ointoff is x y Therefore x y converges to x y

o o k k o o

s u

Therefore x y W x y Similarlyz w W x y

o o o o

Acknowledgements

This work was partially supp orted by NSF grants DMS and DMS

Thank you to Oliver Go o dman for carefully reading a previous

draft of this pap er and making valuable suggestions for improvement

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