View of On the classification of dynamical systems

MATI{EMAîICA

-

REVUE D'ANAI{YSE ^NUMÉRIQUE ÞT ^DE TIIÉORIE DE IT'APPROXIMATION

L'ANALYSE

^NUMERIOUE

ET LA THÉORIE DB

L'APPROXIMATION Tome

B,

No

2'

1979,

PP.99-109

oNTHECLASSIFICATIONoFDYNAMICALSYSTEMS

by

GH. TOADER (Cluj-Napoca)

1.

Infioduction

The set of

dYnamical ^sYstems

problem" is

unsolvable. How-

åittg so-" Partial

^{answels' So} ontäins

a

nãcessary and' suffi-

In thjs

^{papef we pfopose á.methocl}

^similar to

^l:hat

^of

Vrublevskaya

but

basecl dn

-a

simpler

^definition

the

Pompeiu-Hausdorff

metric in

^th'

similar

^dne

in the

case

of the

^GH'

clefined

for the set of

continuous

in

_[11]

and

have used

then in a

_Pr

i

i

¡

100 GII, TOADËR

2. Basic notations and delinitions

I-,et

X be a metric

space

with a fixed metric d,:

^d,*.

Êor

^a.

e X,

r>0 and A (X we

denote

by:

(1)

d'(ø,

A) : ial

_{d,(a, x)

; x ^{e A}}

the

distance

from ø to A

and

by:

(2) V(ø,r) : _{{x e} X;

^d,(a,

x) < r}

the ball of radius r and center a. For a

function

f

^:

X* Y ^{and a}

^set

AUX we

denote:

(3) _Í(A) : {Í(x);

^tc

^{e A}.}

Def inition

1.

A

d,ynømicø|, system on

X'i,s a

continuous funct'ion

n: X X R- X ^that

søti,sfi,es the fol'l'owing ^{øxioms :}

(Ð n(x,0) : x, foÍ every x in ^X;

(ä)

æ(rc(x,l),s)

:

æ(x,t

I ^{s), for} x in X,

ú

and

s

in

R.

Def inition 2. For eny x ^e X

one d'efines:

a) tke

mot'ion

(through x) n,: R-* X by:

f4) ^n,(t) :

æ(x,t);

b) the

trajectory

of x ^by:

(5)

_{[email protected])}

: _{æ(x,t); I ^e

R} i

c)

the þositiue

l,imit

set

of x

^by :

(6) L*(x) : _{{y =} X;

^{Jtn ---+}

| ^æ,

n(x,t,)----y}.

Def inition 3. A _þoint x ^e X

^(and

^its trajectory) is

^søi'd'

to

be:

a)

cri'ticøl',

iÍ y(*) : _{x};

b)

_þeriod,ic

if

^tkere

is ø þ10,

^{such that æ(x,}

^{t +} þ) :

rc(r,t)

for

^øll

teR; c) L

^støbl,e,

iÍ _{[email protected])}

i's rel'atiael'y ^comþact;

d) þ

^{Poisson støble,}

i'f x e L+(x);

e)

e-støbl,e (in

the

sense of M.

BEr

or{rNo

lll) if ^øny

e-ne'ighbourhood'

oÍ [email protected])

contø'í,ns

at

leøst

ø

trajectory d'i'stinct

oÍ

_{[email protected]).}

Def inition 4. Let n

^ønd,

o

be d'ynamical' systems on

X

oel,y

on Y. Tkey

øre NS-'ísomorþkic (respectively GH-isomorphic) ex,ists

ø

homeomorþhdsm

h: X----Y,

^uhích' _þreserues trajectories,

(7)

^y(h(x))

: [email protected]\, ^for all n e X

resþect'i-

if

^there

tkøt is :

3 CLASSIFICATIoN oF DYNAMICAL SYSTEMS 101

(respectively

which

makes commutative

the following

diagram:

X x R

ⁿ

-X

(B)

_lz

^* r. _I,

v i ⁿ _-'--Y

that is:

(B') k(n(x,t)): "(h(x),/), for

^a71

x ^e X ^and

' ^e

^R)'

In what follows we shall ^use the set

1

:

_[0,

1]

^and

the

function

).:[0, æ]*.I

defined bY:

I' Îorte

[0,oo)

(e) ^r(t) :{t+,

I I forl:co

on the set of non-empty

subsets

of x

^we

shall

use

the

premetric

p oi Þoãpeiu-Hausclorff

[-61- definea

for anv M, N C X bv:

(10)

P(M, N) : I(sup {sup {d(ø,N) ; ^{x e M\,}

^sup

{d'(y,M); y _{= N}})'} I,et c(R,X) be the set of all

continuous functions

from R to X.

We

^need sòverál metrices

for

C(R,

X)

^:

a) the uniform metric T,

defined bY:

(11) T(f,e): I

^(sup

{d(Í(t),s(t));t = ^R}), for

anv

Í' ^g = ^C(R' x)

;

b)themetricK,whichgeneratesthecompact-opentopology,

defined

by:

K(Í, Ð :i r-"

^r(max {d(f Ø,e(¿))

;

l¿l

< n\);

the metric S, of

Pompeiu

- ^{Hausdorff type, which we}

^defined

I ^bv:

s(l

^g)

:

tr(sup

{so(l s),

^so(g, _"f)})'

2

(r2)

in _tl

c (13)

) 1

where

(13') S,(l

^g)

:inf _{{r >} 0;Vr ^{e R,} inf {d(f(t),

^e(s))

;

^l¿

-

^sl

{ ^r\ <r}'

with the usual

convention

: ^inf

^Ø

:

æ _'

ft ^is

^easy

^to

^check

^{that for any.f,g} = ^C(R,X):

(14) ^K(f, e) < ^?Lt e) ^{and s(/,}

^s)

^< T(l

_e)

702 GH, ToADER

and, as we ^proved in _[11], the identity

function :

i

: (C(R, X),

S)*

^(C(R, X), K)

is

continuous.

If it is

necessary,

we indicate by a

lower

index the

space used

in the definition of a certain metric (for

exemple

P¡).

L

^{e m}

ma ^2, If _f ^{: X---,Y} is a uniform

continuous

function uith

^the

þroþerty that

f(M)"e"w Íor ^any M e

^(J then the ind,uced, functioø

f :u- -W ^is

continuous.

Proof

. we

proceed

by

contradiction.

Le!

ug.supp-ggg.

that there

exists

u cottvetg"nt

^{sé^quence}

M**M

^such

lhat.f(M") *Í(A), ^{that is}

^{one can}

find a r )

^{O sucñ}

^that

for-any no

thete

exists

n)

tto

for what P"(Í(M")' Í(M))> r. ^{There are two}

possibilities:

a) sup _{d"(f

^(*),

f(MD; ^{x e} M"o)> r;

b) sq {d'(f(M"u), fU))', ! ^{e M}2r,}

for

some

frh*æ. In the first

^case,

^{for any} å

one can

find a

xnhe M'rh such

that:

(15)

dt([email protected]"*),

Í(x)) ^> ,, ^for

^anY

^{,í e M.}

But

/being

^uniformly continuous there exists

a

s

> ^0,

^such

^lhal

^d,v(x,y)

1 ( s'impliãs d't(f(x),Í(Y)) <r,

^and

with the ^property that for

h

)

^hu,

exists

lo = ^M

^wít]¡ d'v(x,*, Yo)

<

^{s, he}

dicts

^(15).

fn the

^case

b) one fall the simple continuity of /.

Consequence ^{1. Let} X

^ønd'

Y

be comþact

mlllt-sþaces,-V o!

W famities

oi.subsets

of X

resþectiael,y

Y. IÍ Í(M) =

^W

for

^ønynM

e

_U

a.nd,

Í-1(N) = ^IJ for ^any N ^e W,

^then

^the

^ind,uced' function

f ^: U"-W

^is

a

homeomorphism.

I,et us äpply

^these

results to

dynamical systems-

For a

dynamical system

tr ott if

^{we denote}

^by læ the family of all

trajectories.

Ifow

^one

kîows [A],

^1.æ

is a ^partition of X

^{and we consider}

^on it,

as before,

the premetric P ^of

Pompeiu

-

Hausdorff and

the

corresponding equiva- lencË

relation which

^we-

^name in this

case 1-equivalence. One obtains

the ^quotient

space

îr, und the

following:

TIrEOREM

L If

^tke d,ynami,cal systems ¡c ^ønd, a d,efined,-on tke comþøct

rnetric sþacets X^rbsþectiael,y

Y

are-NS-isomorþkic, tken the corresþond'ing quoti,ent sþaces

læ

ønd,

lo

øre homeomorþhic.

Rernarh

2. This condition is not sufficient for

Ns-isomorphism how shows

the

following:

ExemþIe

l. Let X:Y : {(x,i) e Rzi x'*y1.5-3}

_?s- ^subspaces

of R2. The dynamical systems defiàed.-by autonomous

differential

equations

(in polar

coordinates) ^:

ä:p(p-1)(p-s);

g:1.

4 5 CL,A.SSIFICATION OF DYNAMICAL SYSTEMS 103

3.

Conditions

lor

NS

-

isomorphism

I.et X and Y be two metric

spaces, and, æ

and o two

dynamical systems

on X

respectively

on Y. For theirs

NS-isomorphism

the

two

families of trajectories (of

æ

and of o) must

correspond. each

to

other

by o

homeomorphisrn

k: X-"Y. ^But the families

being

infinite, it

^is

dilficult to check this

correSpondence.

Thus it is natural to look for criterions

which contains simpler conditions,

at least

necessary,

for

equi- valence.

We begin with

some

lrrore

general considerations.

On

any family of

subsets

of X we

consider

the

premetric

P

defined

by

(10).

I.et U be

such

a

family.

Def inition 5.

We say

that tuo

sets

M,N C{ ^are

U-equivalent

(and

denote

by M - N ^{(rel UD i"f}

^{there exists}

^a

continuous

function h:

I* U

such

thøt

h(0)

: M

ønd,

h(l) : _¡¡.

Remørh, 1.

It ^is

easy

to verify that

U-equivalence

is an

equivalence

relation,

hence

it

^induces

a partition of U. We

denote

the

equivalence class

of M by û _^na ^{the quotient}

space

bv

Û.

I,emma

1.

If U

ønd,

W

are

fømilies of

subsets

of X

resþectiael,y

Y, tken øny

continuous funct'i,on

F: UnW

^d'efines

a

continwous function

^^^^¡A

F:UnW by F(M):F(M).

Proof.

It ^is

^obvious

that M - N ^{(rel U)}

^implies

^F(M) -

^F(.N)(rel

W) so

^Llnat

F is well

defined. Using

the

continuous canonical projections

i:U-U and j:W*W we obtain the following

diagram:

(JF-W

l¿ ^l¡

I ^ l'

Ù ' -û,

F'rom

his commutativity

results

the continuity of Ê b"""or"

₇ ^o

F is

con-

tinuous

_{(see t5l).}

(")

r04 and

(")

p:

GH, TOADER

*: ^x(x - ^l)

!:0

6

CLASSIFICATION OF DYNAMICAL ^SYSTEMS 105

I

Then the

l,agrange

with the

^trajectorY

that is the

theorem

LemmaS'Iftkecløssof.y-equiuølenceoføcriticøl'þoi'ntxisnot a

singleton, them

x is

^e-støbl'e'

The condition is not sufficient as

^shows

the

following:

Exemþle

3. In ^{the dynamical}

^{system generated}

^{by the}

differential

system (iå _Polar

coordinates)

p

sinl _,

^{f.or p}

*0i

P

0 , for P:Q' 0:1;

the origin is an

e-stable

critical point

whose class

of

y-equivalence rs ^a singleton.

4.

Conilitions

for

GH

-

isomorphism

1

because

the GII-

irs

NS-isomorPhism

[13]). But

^{we look} For a dynamical system ^7r on

X,leL us

consider

the

space

of

motions ^:

fI:{zcn; xeX)

and the

maP

æ*:X* [I

defined bY:

(16)

^æ*(x)

:

nn

The

use

of the metric T ^f.or fI

Pro

two ^{motions on the}

same trajecto

the following:

Exemþle

4. Let

^æ

be a

dynamical system

on R definedby n(x't):

:

rc

' exp (t).

1L

x* _Y ^we

^have:

T(æ',

nv):

tr(sup

{lr - ^{yl exp (t);} t ^{e R})} :

1'

stable

traiectory {(x,l); ⁰ < x < ^{1} is}

'¡-equivaleut

{(x.O\'. d < ^x < i} ^{wtrich is not}

^{l,agrange staole'}

là-

;å¿ true if

one renollnces

at

completness' t

p(l-p) if0<p(1;

0 if ¹ < ^p ( 2;

(p-2)(p-3) íf 2<p<3;

0:1

afe not Ns-isomorphic although the quotient spaces l1rc

and Ilo

are homeo- morphic.

As

concerns

the

equivalence classes, we have

the

following:

the systenr,

^ed,

^{on a}

comþIet,loc.øl'Iy

X,

^ais

of

^ce

^of

^{ø Løgrønge støbl'e}

nly

^stable

Proof.

Let

To

be

aÍ"agtange stable,trajectory, 11

:

^{Y0. and}

^h: I*læ

continuoús

and,'ðuch thaí nP\ :

_"¡o,

h(l): _Yr:

^D-eno-te

lv I : ^{t-=*!

^|

nftf i,

Lagrange stable).

of

còúrse 6".=

i,'thai ^is ¡+Ø' ^Let

^to

^e J' T¡.

space

X ^{being locally}

^compact

^and ffi

^{c:gmpact, k,(to) has}

^a

^compact

"ãi*rr¡å"tr'àãål ur"l

^iå

₇ ir

ãpen

in /. ^Èdî /

^{is-also closed}

^{in 1.}

^rndeed,

ii;,;";;pp;."'tir" "onúrty,'thcle ^{exists a}

^{sequence (1,)}

in / ^which

^has

a limit point

to

ê J.

^That

sequence

(ø,) which has

^no

hai'e also k(t,)-

^71¡0]r

^{(!n th}

the set of

subsets

of X). If

bers,

then for

everY

þ,

^thete

^{is a}

ⁿ

implies P(k(t^),h(to))

<!'t'.tm1)Ntanð'yl, =îñsuchthat

^d(y"'

^{x,) <}

< 3 for ^afly n.

^Because

TAÃ is

compact, there

is a

subsequence ^(yl,o)

of ^{(yt-) which is}

convergent.

We may

assume

that a0'"0,

y|r)

< 3

for any

h

^anð.

l.

^Denoting

^{x,o tty}

^x'0,

t'. ^{obtain the}

subsequence

(øl) oÍ

(x*) such

that

d,(xte,

x!) (

e,

for

any

h

and

l'.

Step

by

step,

for þ :2, 3,

_{' ' '}

we obtain the

sequence @!,)

such that d(x!, xt) (

e¡

for

^any

h

^{anð' l,}

(x!"\ being

subsequence

of lxl-'). ^{So çxi) is} a

Cauchy. sequence. and,

Ï"f"itrg--Ëo-piãi=it ^is "

convärgãnt subiequ_"ttc".

of (1,) in

contradiction

*itrr--iË" urän*þtiou- Att"t uil, _J : I, ^{^that} is

^¡f

is

l,agrange stable.

ExemþIe

2. Let ^X :

R2

- ^{{(0,0)} with the} usual Euclidean

^metric and. rc

thå

dynamical system gèierated

by the differential

^system:

p:

t

106 ^{GH. TOADER B}

I

CLASSIFICATION OF DYNAMICAL SYSTEMS ¹⁰⁷

*xo

such

that is no motion

is. equivalent

with

another

(taking the metric T

^for

the

space

of

motions), ^-

ln what follows we shall

^use

avoid

these

difficulties. With this

c

D

ef initio n

G. We say that

Ient (and nII,

suclr denote

thøt

h(0)

by : ¡- - ^n,,\'if

^the

".

;;Nd"h(l)

As

above

we

denote

the

class

of

Il-equivalence

of

nn

by i

^{and the}

quotient

_space

II/_by û.

L

e m

tna 4. If

y(x)

: y(y), then

nn

-

^ær.

_ ?o-tf.By ^the

hypothesis

!:n(x, ^{s) with}

^some

s e R. Definig å:

/*II by

h(t)

:

æn1*,¿"¡

it is

continuous because

s(h(tr),

h(tr))

< lr, _ t,l ^.

_lsl

and _{Remark k(0)}

:-rr, _2.

_Thus

_þ(l) : _s iry, that is n, - ^v*

is

more usefirl

thdn the metric T. Also,

generaily

it

induces

not trivial

quotient. _spaces..

rot

"""-pìã,-"t tt"'dînamicat

system

..appearing - in

. exemple

4, óne obtain three'II

jequivalence

"1"r;;;

(upon the sign

of. x).

ular

classes

of

motions.

First of

all

definsd Iw j(") : _{[email protected]), is}

_conti-

zur)..

Thus every class of

fl-equi- -equivalelce. _So,

for

ll-equivaleäce

contained in the

theorem

2

and

L

e m m

a 5. The set of

motions þosi,tiaety

poisson

stabr,e ,ís crosed,

in II.

Proof.

r'et

the-

motions

æn,

positivery

^poisson

stable and

æn such

t!^l S(",,, æ,)*0 fot n-co.

TËus,

for any natural þ

^there

^{is a øl}

^such

that:

(17) [email protected],,,,n.¡ q

i.e. for soÍre s¿,

_lsel

< ^{ll3þ we}

^lnav

fi

being positively

^poisson

stable

so

1

^such

that:

d,(x,

n(x,

"¡)) < ^llþ

By the

theorem

of poincaré-B"náixon

[10]

^{we have}

the

following:

Conseçluence ^{2. For a}

d'ynømi,cø|, system, on tke þlane, ^{the set of} þeriod.ic rnot'ions

is

^closed'

in

^IL

Def initíon7.

The function

f :R*X

^has e-þeriod"c

if for

^øny,

t ^{e R, d(Í(t} + r), f(t)) <

^e.

L

^{e m m}

a 6. If tke

cløss

of lI- ^e of a.

^{motion æ*0,}

ot

þass

^{xo, then}

^for

ry

^d,o

n ⁿ

^{xo, uhiclt,}

{e. is uniformly

continuous

on _[-1,

,

1!, 4l'

_such

_that for any t,s = ^[-1, r + ll, ^with

^l,

-

^sl

^{< 48, we}

^have:

(18)

^d'(æ(xo,

^t),

^n(xo,

^t)) _{< å}

^.

Ry

hypothesis, there

is a x ^e X

such

that

S(æ,., Íç,)

I I

^{and' æ,(l)}

lir anï teR. ^Thus for any t eR, ^there is øs¡, lsr-ll ^<28

that

d,(æ(x,t), æ(xo, ^s

)) < ^{2S' So we have}

succesively:

ls,+"

-

^(r,

*")l(

^ls,+,

-

^(t

^+t)l *

^ls,

- ^{rl <}

^4ò

and,

denoting s¿¡"

-

^na

^{+ r0, with lo} e _[0, r),

^n,

^e Z:

ls,+"

- ^nr -

^(s,

f

^c

- ^nr)l <

^48,

hence

^s¿+t

- ^tt'r,

^st

* ^c(l - ⁿ⁾ = l-1, t f l]

antl by

^{(18) :}

d,(n(xo, s¿¡,), nl()to, s,))

:

d'(rc(xo, Sr+"

- ^nt),

^{n(xo, s,}

* ^(l - ^n)")) < ^tl7'

Finally:

d'(n(x,t

I r),n(x,t)) 4

d(æ(x,t

I

^{r),n(xo, s,+'))}

*

d'(æ(xo, s¿1"), æ(ø0, s,))-¡

I

^d,(rc(xo, s,), æ(x,r))

<

^2S

+ ; ⁺

²⁸

(

e,

thus

æ, satisfies

all the

expected' conditions,

Lem ma 7. If ^{the møþ} h:X_*Y is uniformly

coøtinuous, then ^so

is

al,so

tke maþ

h* : C(R,

X)*

C(R,

Y)

^d'efined

^bv:

(re) (h.(Í))(t) :

h(f ^(t))

(using

the metric S for the

spaces

of

continuous functions)'

Proof.

For any

^e

) ^{0 there is a..ô, 0} <-l I

^e,

^with

^{tEe. property}

that d*("x,!) ( ô ^implies

^d,

^(k(x),

^nOD

<e.

^Tf.

_f,g = ^C(R,X)

^{are such}

108 CH, TOADER

XxR n

-X t*

hx l* -

YxR

^o

that S*(f,g) <

^8, th.en f91. any

t ^e R there is a

s,

e _{R such} that

_ls, ^_-

- ^tl < I and d*U(fl,

_S(s,))

< ^à.

Hence:

min

_{d.r(k(f

(t)), h(g(s)));

_ls

- ^rl { ^{e}ç min}

_{d,r(k(f ^(t)),

[email protected]));

_ls

_ _rl <

^s)

<

dy(h(Í(t)), _h(s(s,)))

<

^e.

Changing

the role of f

^and

^g

^we

^get

^SyØ*(f), h*(e))

<

".

THEoREM S.

Let îc

^qnd,

o be

d,ttnamical, systems on comþøct s7laces

-xorcsþect'iuetv

y. _rr

_thev

_are cti.-d'"*lïph;;,";tu"; ííí

quoil,rnt

fI

ønd,

2 of

cløsscs

of

equiuølent motions are homeomorþhic.

Proof. We have the following

_diagram:

10

metric

sþaces

11 ÕLÄSSIFICATION OF DYNAMICAL SYSTEMS

Rernarh

5. The condition from

^theorem

3 is actually

stronger

that of

theorem

1, and a

class

of

lI-equivalence

may be

^properly

tained." in a

class

of

y-equivalence, as shows

the

following:

Exemþtre

6. I,et o be

defined

as

above

by:

(o) ff:o p(l

l'o: ^op'

109

than ,rcon-

One obtaine

a

single class

of

^¡-equivalence

but two

motions

on

different

trajectories are not fl-equivalent. Thus o is

NS-isomorphic

with _!h"

dynamical system æ from

exemple

5 but is not GH-isomorphic with it.

h*

T

I J

'-fi

.l

h II

Y

I lî

t

REFERENCDS

[1]

Bertolino,

^{M,, Solutions of} ilifferential equalions ì,n arbàtrøryt neighbourhoods of giuen functions (Setbo-Croatin). Mat. Vestnik, 13 (21Ì),

l, 2l-33

^(1976).

[2]Bhatia, ^N,P., Franklin, M.L.,

Dynørniaalstlsternsuithoutseþaratrices. tr'unkc.

Ekv., 16, 1-12 ^(1972).

i3l Bhatia,

N. P., SzeCö, G.P., Stabi,Iilytheory of dynørnical s)lstems. Spriuger-Verlag, New York, Heiclelberg, Berlin, 1970.

t4l ^{H ó j e}

k,

O., Cølegorial conceþis in dynømicat syslem lheory. fn ,,Topological Dynamics", J. ^Auslander antl W.

II.

Gottschalk (Editors), New York

-

Amstertlam: Benja- min, 1968, 243-258.

[5] ^{K e l l e}

y,

J. L.,General, ^Toþology. D. Van Nostrantl Company, fnc. Princeton, New Jersey, 1957.

[6]

Kuratowski, K.,

Toþolog3t,

I-ff.

Academic Ptess, New York and, I,ondon, 1968.

iZl

^{tvt a ¡ k u s,}

\,,

^{Globøt structuri oJ} ordinarlt differential equations in ^the þlane. Trans. AMS, 76, 127-148 (1954').

[8] ^{M a r k u s, L., Lectuyes} in DliÍfeyentiable Dynamics. Regional Conference Series in ^IVIathe-

matics, suppotted by National Science Foundation, Number 3r 1971.

t9l ^{M a r k u} s, I,., Porøltet' d'ynømioal' systems. Topology, B, 47 -57 ^(1969).

itÓl

¡qemytskii,

V.V.,

Stepanov,

V.V.,Quali'tatiueTheory oJ Diflerentiatr Equations'

[1]

^{T o a}

d

^onsi

10t1,

tttl orie

L^-l

[13] ^U r

a,

their isomorþhisms' Japan

-

U.S. Sem. OrtI' Diff.

cture Notes l!'l:at'ln., 243, Springer Verlag, ^1971, 76-90.

[I4]

Vrublevskaya, I.

N., Oz ^geometric equiuølenae of the trajeoloriesof ilynamiaal systems (Russian). Mat. Sbornik, 42. 361-424, (1957).

Received s'vrr'

1927'

cenrrul leritoriøl ^dø aalaul eleatrcnic Sh. Reþublicii 107 3 400 Cluj-Nøþocø o*

where

the first

rectangle

is from the

GH-isomorphism

of

æ

and o,

æ*

and o*

are defined

rv _{[te¡, k*} b; (rE,

¿

""Jitãi"'"Jnäni""l

projections

irr quotient

spaces anð.

h is

defined

by:

î1î"¡ :

Ç,o,

From the continuity. or h*, one

deduces (sge

rs]) the continuity of î.

Fr.tt"ttn ^{the proof} it is

eriough

iJì"p"ut'ttre äuäve-"onsia"i"iió"r-ráï

Remarh,

3. v/e

cannot use

the commutativity of the last two

rectan_

gles

and avoid

_so

the proof of ilt"-"ã"tinuilv _ái ãi,'îãäo."

rc*

and

o*

may be

discontinuous.

Remarh,

4. As

in

-

the

case

of

NS_isomorphism, theorem

S is not sufficient

_for ^- GÈ_Iso-orpfil.rr¡--'

wing:

Exemþle

5' Let

æ

an. o

be-.dynamicar systems defined

by the

dif-

fererrtial

dystems

(in polar Ã""øi"áiár- - -'

(") _lþ:o

tb:t ^(P<l)

the

condition from how shows

the

follo-

(") p:0

b:P ^{(p< l)}

I I

The

systems

are Ns-is _morphic but not

GH-isomorphic, although _the

quotient

_spaces

fi and î ur"

homeomorphic being singretons.