ISOLATED INVARIAN T SET S

5

portraits contai n (respectively ) one , four an d twelv e nonempt y isolate d invarian t sets . Fo r

example i n th e middl e portrai t ther e ar e two set s consisting o f a single rest point ; one o f th e

interval connectin g th e point s an d on e consistin g o f bot h res t points . I n th e botto m portrai t

there ar e three on e poin t sets ; three consistin g of a n interval; three tw o poin t sets ; two con-

taining on e interval an d a point an d on e with thre e points . T o some exten t th e "complexity "

of th e equatio n i s measured b y th e numbe r o f isolated invarian t set s an d th e relation s be-

tween them .

Choosing N t o b e the interva l indicate d i n th e to p portrai t (o r mor e precisel y it s corre-

spondent i n the rea l line o n whic h al l the equation s ar e defined ) i t i s seen tha t th e left-han d

rest point s in each o f th e thre e portrait s ar e relate d b y continuation : imaginin g th e interval

to b e translated verticall y downward , on e find s i t i s an isolating neighborhood fo r ever y equa-

tion correspondin g t o a value o f X between thos e o f th e to p an d botto m portraits , an d it

isolates th e left-hand res t poin t i n every case.

Similarly, choosing N a s indicated in th e botto m portrai t (an d translatin g it verticall y

upward) it i s seen tha t th e res t poin t i n th e to p portrai t i s related b y continuatio n t o th e ful l

set o f bounde d solution s in eac h o f th e othe r tw o portraits. Becaus e the relatio n ha s been

made a n equivalence relation , the abov e remark s imply tha t th e left-hand res t point i n th e

bottom portrai t i s related b y continuation t o th e ful l se t o f bounde d solution s i n tha t sam e

portrait. I n a similar wa y (going into th e lowe r half-plane) i t is seen that th e right-han d res t

point i n th e botto m portrai t i s in thi s same equivalence clas s o f isolated invarian t sets .

However, not al l the isolate d invariant set s are in this class ; for example , the middl e

rest poin t i n th e botto m portrai t i s not, since no choic e o f N wil l continue thi s set up t o o r

beyond th e middl e portrait . O f course it i s conceivable tha t th e give n family o f equation s

might b e embedde d in a larger on e in suc h a way tha t th e middl e res t poin t continue s t o th e

rest poin t o f th e to p portrait ; however i n thi s simple example thi s is easily rule d out . I n

general, however, it i s convenient t o have a (hopefully computable ) invarian t whic h wil l dis-

tinguish (some of ) th e equivalenc e classes .

3. Th e Morse index.

3.1. Th e Morse index o f a n isolated invarian t se t (a generalizatio n o f th e Mors e index

of a nondegenerate critica l poin t o f a gradient flow ) i s an invariant o f th e equivalenc e classe s

of isolated invarian t sets , and it carrie s som e information abou t how solution s near th e in-

variant se t behav e near tha t set .

This index take s th e for m o f th e homotopy typ e o f a pointed topologica l space . A

pointed spac e means a pair consistin g o f a topological spac e an d a distinguished poin t i n tha t

space. Tw o such spaces are homotopic i f the y ca n b e deforme d eac h to th e othe r in a way

which respect s th e distinguishe d points . Th e equivalenc e classe s s o define d ar e calle d homot-

opy types .

3.2. Computation of the index. Indice s ar e compute d fro m specia l isolatin g neighbor -

hoods calle d isolating blocks. I n th e exampl e o f Figur e 1, any isolatin g neighborhood i s also

a block: roughly , th e definin g propert y i s that th e solutio n throug h eac h boundar y poin t o f

a block goe s immediately ou t o f th e bloc k in on e o r th e othe r tim e direction . Thos e bound -

ary point s whic h leave a s time increases ar e calle d exi t points . Th e index i s the homotop y