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A simple-minded approach to the index of periodic orbits
JOURNAL
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
129, 517-532 (1988)
A Simple-Minded Approach to the Index of Periodic Orbits CHRISTIAN
C. FENSKE
Mathematisches Institut, Justus-Liehig-Univer.vitiit. Arndtstrasse 2, Dd300 Giessen, West German? Submitted by Ky Fan
Recevied July 17, 1986
We define an index for periodic orbits of semiflows which satisfy a mild compactness condition on infinite dimensional ANR spaces without requiring differentiability. The index is a one-dimensional homology class. I( 19XX Academ,c Presr, Inc
1. INTRODUCTION In [2], F. B. Fuller defined an index for periodic solutions of autonomous differential equations on manifolds. In fact, his article describes several indices and we shall concentrate here on the approach which defines the index of a periodic solution as a one-dimensional integral homology class. Fuller’s approach used cohomology of differential forms, so he had natural assumptions on differentiability whereas we intend to deal with the continuous case. So we do not consider periodic solutions to a differential equation but rather periodic orbits of a flow-or, more generally, a semiflow since we shall be mostly interested in an infinite dimensional situation. A semiflow on a topological space X is a continuous mapping @: Xx [0, co) -+ X such that c$,,= id and dffs = d,#, for s, t 2 0 where we have written ~,4,=@(., t) for 220. Semiflows naturally occur in the study of retarded differential equations and it is there that one is interested in getting rid of differentiability assumptions on the semiflow. As to the underlying phase space X we shall assume that X is an ANR and we refer the reader to [3] for material concerning ANR spaces.We will start by considering a situation where X is open in a euclidean space, then we generalize to simplical complexes, and finally we approximate an ANR by nerves of open coverings. The index we shall define will be additive, homotopy invariant, and in case of a single periodic orbit the index can be described in terms of (the classical fixed point index of) the Poincart map (which requires some extra proviso in the case of semiflows). 517 0022-247X/88 $3.00 CopyrIght m(, I988 by Academic Press, Inc All rights of reproductmn I” any form reserved
518
CHRISTIAN
C. FENSKE
2. THE SETUP Let X be an ANR and @ a semiflow on X such that - there is a to B 0 such that $,, is locally compact (i.e., each x E X has a neighborhood I/ such that cl d,,(V) is compact) - @ has a compact attractor (i.e., there is a compact set A such that for each neighborhood W of A and each x E X there is a t with dtx E W). Let UcXx [0, 00) be open and call P:= {(x, t)Ecl Ul$,x=x}. Assume that pr,(P) is a bounded subset of (to, co) (where pr, is, of course, the projection onto the second factor) and that P c U. Then P is in fact compact: according to [ 1] there is a compact attractor A for @ which has arbitrarily small neighborhoods V such that @(I’ x [0, co)) c I/ and cl q4,,(V) is a compact subset of V. It is obvious that all periodic points of @ must be contained in A, so P is compact. We shall say that there is defined an index of CDwith respect to U if there is an element Z(X, @, U) E Zf, U (the first homology group of U) such that (A) the index is additive: if P = P, u P,, P, c U1, P, c U, are closed, U,, U, are disjoint open sets with U, u U2 c U then 4X, @, U) = ileZ(X, @, U,) + &,Z(X 0, UJ, where i, : U, + U, i,: U2 + U are inclusions. (H) the index is homotopy-invariant: Assume that @:Xx[0,co)x[0,1]-+X x, t, s -+ @qx, t) is continuous and that each @” is a semiflow. Define Y: (Xx [0, 11) x [0, co) + Xx [0, l] by Y(x, S, t) = (@“(x, t), S) and assumethat Y is locally compact for some to 2 0 and has a compact attractor. Let UC Xx [0, co)x [0, l] be open and call P= {(x, t,S)Ecl U(@“(x, t)=x}. Assume that P c U and that prz(P) is a bounded subset of (to, co). Then i,*Z(X Go, U,) = i,.W,
@‘, U,),
where
Uo=(Xx[O,co)x{O})nU, U,=(Xx[O,co)x(l})nU, and i,: Uo+ U, i,: U, + U are inclusions. (N) the index is normalized: If P consists of a single periodic orbit C
then Z(X, @, U) = i(@“)[C],
INDEX
OF PERIODIC
519
ORBITS
where i(@‘) is the index of the Poincart map associated with C and [C] is the class of the singular l-cycle represented by C in U. (To be precise, let T be the minimal period of the orbit corresponding to C, then [C] is represented by [0, l] + U, t -+ @,,x for some x on C. For the definition of i(Qp”) we refer to [ 1 or 43, see also the proof of (N) in Section 3 below for a discussion of the relevant facts.)
3. THE
DEFINITION
The definition will be given in several steps. We start with the following situation: D = X is open in R”, U is open in D x [0, co), @:U -+ D is continuous, and P= {(x, t) E UI 4,~ =x} is a compact set. Call t the orientation r E H"(D x D, D x D\A), define g: U -+ D x D by g(x, t) = (x, 4,x), and define Z(D, @, U) to be the image of t under
p(DxD,DxD\A)
,,R"(U,U\P)zHfP~H,U,
where R* denotes Alexander-Spanier cohomology, z is Poincare duality [S, p. 3631, and i: P -+ U is the conclusion. We remark: (1) In the situation described above, let P c Q c U, Q compact. Then we may replace P by Q in the definition of Z(D, @, U):
JP(DxD,DxD\A)-+R”(U, u\Q)rHfQ-H,u
‘\,I
I/
A"(U, U\P)zH;P
where the vertical arrows are induced by inclusions (since Poincare duality is natural with respect to inclusions). (2) If V is an open subset of U sucht that P is a compact subset of V then Z(D, @, U) = j,Z(D, @, V), where j: V + U is the inclusion. This is obvious since i, : H; P -+ H, U factors through H, V. Property (A) is obvious if U= U, u U,. For the general case, use (2) with V= UI u Uz. As to (H), we prove this without using the assumption that the homotopy occurs through semiflows. For s E [0, l] let P, = {(x, t, s) E U 1@‘(x, t) = x} and choose a 6, > 0 and a compact set C, c X such that P,cC,x {t} c U as long as Is- tI d 6,. Choose an open set If7c Xx [0, co) such that
P,x([s-6,,s+6,]n
[O, l])c
lfyx [s-d5,s+d,]c
U.
520
CHRISTIAN
C. FENSKE
Denote by j, the inclusion j-V: V, x ([s - 6,, s + S,] n [0, 11) + U. We then have
io,Z(X,CD',Uo)=io,joeZ(X,@O,Vo)=ilJl*Z(X, @', Vl)=il*(X @', Ul) since by (1) we may replace P, by C, in the definition of Z(X, @j’,U) as long as Is - t) d 6,Y.We then choose a finite subcovering of [0, 1) and proceed by induction, for it is obvious that t --f Z(X, @‘, V.,x [s - 6,, s + d,]), does not depend on t as long as (s - tl < 6,. We now turn to the normalization property: We use the notation adopted in (N). We may choose U so small that P is the only periodic orbit in cl U and such that U is homeomorphic to C x B, where B is a small open ball in Iw”. There is a unique p > 0 such that P = { (dtxo, p) 10 < t < p} for some x0 E C. Choose a section S for C. This means that x0 E S and that there is a neighborhood V, of C in U, := Un ([w” x {p}) such that for each xE:C:=Sn v, the map defined by r(x):=inf(t>Ol4,x~S} is continuous, that 4,~ E U, whenever )’ E C and 0 d t < r(v), and that for each y E V, the number q(y) = inf( t > 0 14,y E S} is defined and q 1V\Z is continuous (cf. [ 11 for details). Moreover, we may (and shall) choose V. so small that jr(x) - r(x,)l < r(x,)/8 for XE V,. If @ were a flow it would be easy to show that Z is an ANR, but though a semiflow on D (or, more generally, a finite dimensional manifold) cannot have startpoints it does not necessarily have backward unicity, so the obvious neighborhood retraction onto Z along flowlines does not work. We therefore use a substitute for the Poincart map which will be defined in a neighborhood of C. Geometrically, think of @ as flowing from the left into C and leaving it to the right, i.e., define V, = (x E V, 1q(x) < z(x,)/4}, V, = {x E V, I q(x) > 3~(x,)/4}. We now speed up @ in V, a little bit and slow it down on V,. This will have the effect that x0 is the only fixed point of the modified @ in V,u V, and we use its index as the index of the Poincart map. We use a homotopy which does not affect @ on t3U to deform @ in a small neighborhood of x0 in the same way. Note that during all deformations no point leaves its flowline so that fixed or periodic points can occur only on C (but possibly for other values of the time parameter). The details are as follows: Call 0 := int(c1 V,u cl V,). Choose an E> 0 such that U, x (p - 28, p + 2) c U and define for x E cl V, if r(x,)/8 d V(X) $ $x,)/4 /l(x) := 7(-xo)/4-ul(x) if q(x) G 7(x,)/8 v(x) i and for x E cl V, if xES if 37(x0)/4 < q(x) d %(x,)/8 if 7~(x,)/8
INDEX
OF PERIODIC
Then extend ,U continuously over {x~cl V,I q(x)> 7$x,)/8} cl(X) E C-~(X,Y8,01. ff xEc1 6 and It-pi 622~ let
;I(x, t) := o(E- It - PI 1P(X)/&
521
ORBITS
such that
if It-Z7
Now we define a deformation h: cl V, x [p - 2c, p + 2E] x [0, l] + D connecting h(x, t, 0) = dtx with F(x, t) := h(x, t, 1) by h(x, t, ct) := 411(.Y,r)+rx i 4 ,X
if Odd(x,C)<2E and It-pi 62~ if d(x, E) 3 2~ or d(x, L’) < 2~ and c6 It-p1
d2E.
By definition, the in&x of the PoincarP map, i(@“), then is i(@“) := ind(D, F( ., p), 0) where ind is the classical fixed point index. Note that by (H) we have that Z(D,@,
v,X(p-2E,p+2E))=Z(D,F,
V,X(p-2E,p+2E)).
Now embed V,, into V via i(x) := (x, p) and call Z:= {(x, t)~ VI F(x, t) =x}. Then Z is a compact subset of V (and in fact homeomorphic to a one-sphere). Choose the obvious generator y E H, Zz H, C and call c the generator of I?( V, V\f) which is mapped onto y under the isomorphism Z?‘( V, V\Z) r H, ZY Write Z(D, @, V) = Z(D, F, V) = I. y and denote by t the orientation r E R”(D x D, D x D\d). So r is mapped onto I. c by the homomorphism l?(D x D, D x D\d) + R”( V, V\Z) which is induced by (x, t) -+ (x, F(x, t)). Now if n 3 3 we have that (with j: 0 + V being the inclusion)
H”(v, v\q
.I* * H”(O,~\{x,})
II I’H”-‘(~\{x”}) I ijn-‘(V\r) and all groups are isomorphic to ;2. Co\{x0} is homotopy equivalent to S”- ’ and V\Z is homotopy equivalent to S’ x S”-’ and it is easy to see that j* maps the generator of
onto the generator of H” ~ ‘(S\{ x,}) 2 H” ~ IS”- ‘. (A similar argument works if IZ= 2. We then have H2( V\Z) E Z @Z and an isomorphism H2( V, V\T) E H2(0, O\{x,)). But this means that the index of the Poin409,129’2-I2
522
CHRISTIAN
C. FENSKE
care map, which by definition is just the fixed point index of F( ., p), equals Z since the fixed point index of F( ., p) = Fo i is the image of r under R”(DxD,DxD\A)------+
WY
v\o
i*
fz”(G,
c?\{X”}).
This completes the proof of (N). At first glance it might seem that one could prove the normalization property (N) just by noting that Z(D, @, V) = J. [C] where J is the fixed point index of the map V+ D x [0, cc) given by (x, t) + (4,x, p). This does, however, not work-simply because it is false. The point is that we obviously have J=O; just reparametrize @ by a homotopy so that the periodic orbit occurs at a time parameter different from p. In addition, we note that instead of [C] we might have taken the class of t + d,pxO (whereas above we have used the class of t + 4,,x,). This would have to be compensated for by dividing i(@“) by p/r. We now turn to the case where X= K is a (finite) simplicial complex, @ is a semiflow on K, U is open in Kx [0, cc), and P= {(x, t)E Ulcj,x=x} is compact. We choose a PL-embedding h: K -+ R” into some R”, an open set D=J K, and a retraction r: D +/z(K). Let R :=r x id, H :=h x id, and define g: R-‘(H( U)) -+ D x D by g(x, t) := (x, h#,h-‘rx). We then define I,,, .(K, @, U) to be the image of the orientation TE R”(D x D, ~;~ZI~/)_~;der ~!“(DxD,DxD\~)--@*R~(R-~(U),R-‘(U)\H(P))E c * H, U. We are then left with the tedious job of showing that Zh,lr(K, @, U) does neither depend on the embedding nor on the retraction so that we may put Z(K, @, b’) := I,,, ,(K, @, U). We start by showing that the index does not depend on the retraction. In order to simplify our notation let us assume that K is already a subset of R”. First, it is obvious that we may replace D by a smaller neighborhood D’ of K and r by r 1D’. Let then rl: D, -+ K, r2: D, + K be retractions. By (2) we may replace D,, D, by D := D, n D,. Define q: (Dx {O})u (Kx[O, l])u(Dx {l})-+K by q(x,O)=r,x, q(x, t)=x, q(x, l)=rzx. Then we find an open neighborhood D’ of K and an extension Q of q over D’ x [IO, 1). Again, we may replace D by D’. So we are reduced to the situation where we have homotopic retractions r, , r2 : D -+ K and (H) shows independence. Now assume that K is PL-embedded into R”, that m > n, that D c iw”’ is a neighborhood of K x (0) in R”, and r: D + K is a retraction. As we have just seen we may assume that D = D, x W where D, is open in R”, W is a neighborhood of 0 in R”~“, and r = (r,p, 0), where p: R” + R” is the orthogonal projection and ro: Do + K is a retraction. Write points in R”’ as (x, y), XE R”, YE lR”-“. Then Z,(K, @, U) is by definition the image of the orientation
INDEX
OF PERIODIC
523
ORBITS
z,=T,,xT,ERm((DoxD,,DoXD,\A)x(Wx zA”(D,xD,,D,xD,\A)@~“-“(Wx
w, wx W\A)) W, Wx W\A)
which is mapped into B”(R-‘(U),
R-‘(U)\P)rET”(R,‘(U),
R,‘(U)\P)@P+“(W,
W\(O))
zHf(P)@Hg({O))EH;(P) where, of course, R = r x id, R, = r,, x id. This shows that Z,(K, @, U) = Z,W, @, U).
Before we show independence of the embedding we prove a weak form of the “commutativity property.” LEMMA 1. Let K be a simplicial complex in [w”, L a subcomplex of K. Let @ be a semifrow on K, UC K x [0, 00) open and assume that is compact and that C$,XEL whenever (x, t) E U. P= {(x, t)E Ul&x=x} Denote by i: {(x, t) E ZJ x E L} + U the inclusion. Then Z(K, @, U) = i,Z(L, @, i-‘(U)).
Prooj Choose a neighborhood D of K in R” and a retraction r: D + K. Let R = r x id. Let dp be a neighborhood of L in K such that there is a deformation retraction q: B + L and let Q = q x id. Then r’(9) is open in R” and we may use qr: r ~ ‘(9) + L as a retraction for L. Then Z(L, @, i- l(U)) may be defined using the map @iQR: R-‘(-Y x [0, co))n Q-‘(Un (Lx [0, co)) -+ L. But iQ is homotopic to the identity, so (H) Q.E.D. shows that the index of @iQR equals the index of @R.
Finally, we show independence of the embedding. In order to simplify our notation we assume that K is a simplical complex in IR” and that h: K + K’ is a PL-homeomorphism. By what we have seen above we may assume that K and K’ are subsets of the same R”. Let U be as above, H:=hxid, U’=H(U).Deline Y:Rx[O,oo)+K’by$,y:=hd,h-‘y.We then have to show that Z(K’, Y, U’)= H,Z(K, @, U).
This is achieved by mimicking the proof of the commutativity property for the classical fixed point index. We start by replacing U by a smaller open set so that we may assume dlx # x whenever (x, t) E au. Choose open sets D, D’ in R” and retractions r: D -+ K, r’: D’ + K’. Define a ((2n + 1)-dimensional) open set Q := ((4 y, t)I (rx, t) E U and (r’y, t) E U’ )
524
CHRISTIAN
C. FENSKE
and a map by /l(x, I’, t) = (4,h- ‘r’!:, hrx).
A:KxK’x[O,x~)+KxK
We then define a homotopy l? .Q x [0, l] + K x R” by fJx, J‘, t) := (d,h ‘r’y, xhd,h ‘Y’Y + (1 - c() hrx). We claim that Z-Jx, ~1,t) # (x, y) if (x, y, t) E 8Q and LXE [0, 11. Indeed, assume that T,(x, I’, t) = (x, ~1) then ~EK hence rx=.x and x=cj,h ‘r’y, so hx=zhx+(l-r)hx=.t’. So we have that y E K’ hence r’y = y. This yields #,x=x, consequently (.v, t) $ dZJ, but then also (y, t) 4 au’ since 4’ = hx. So (x, y, t) 4 6X2. In particular, we have seen that f,(x, y, t)=(x, y)for (x, y, z)ESL ifand only if (x, ~)EP:= ((x, t)~ U/dIx=.x} and y=hx. In th e same way we see that T,(x, y, t) = (x, ~1)for x E IF!“, (y, t) E U’, implies (x, t) E U. Denote byj, : Q -+ IF!”x U’ the inclusion. Using (2) we then obtain ,j,*Z(Kx
K’, A,Q)=j,*l(KxK’,
f,,Q)=Z(R”xK’,
f,,
Choose a point x,, E K and define a homotopy R” x K’ by B,(x,
I’, t) := (( 1 -r)
LYx U’).
B: R” x U’ x [0, 1] -+
/,d,h ‘r’y + c~x~, h#,h
‘r’y).
We claim that (x, ~1)# B,(x, ,v, t) if (I’, t) E au’, C(E[0, 11. Assume that B,(x, I’, t) = (x, ~1) then YE K’, so r’y = J and hcj,hk’y = JJ which is impossible if (y, t) E au’. So I(IW”xK’,~,,[W”xU’)=Z([W”xK’,
B,,WxU’).
Now B, is a product of a constant map and Y, so we obtain I(R”xK’,
B,, Wx
U’)=e@I(K’,
Y, U’).
where e is the generator of H,,R”. On the other hand, consider the homotopy 0: .Q x [0, l] + R” x K which is given by 0,(x, y, t) = (coj,rx+ (1 -z) 4thk’r’y, hrx). Again, we claim that 0,(x, I’, t) # (x, y) for (x, y, ~)ELMX, E E [0, 11. Indeed, if 0,(x, I’, t) = (x, ~7) then y E K’ hence r’jl= J’ and hrx = y. So 4,h -‘y = xd,h-‘y+(l-z)4,/1 ‘~=~,rx=x.ButthenxEKandrx=.x,sowehave 4,x=x, hence (x, t) $ iiU, and hx = y, hence (J, t) 4 8U’. Denote by i, the inclusion i, : Q -+ CJx R”. As above, we have that
525
INDEX OF PERIODIC ORBITS
Finally, we consider the homotopy Z:UxKY’x[O,l]~KxR”whichis given by &J-x, t, 4’) = (d,vx, (1 - c() hrx + CCX~). It is immediate Z,(x, t, y) # (x, I’) if (x, t) E au, and we have that I(Kx
R’“, E,, Ux R”)=Z(K,
that
CD,V)@e.
Putting things together we see that j,*I(Kx
K’, A, Q)=e@Z(K’,
Y, u’)
and i,*I(KxK’,A,Q)=I(K,@,
U)@e.
Now define ‘1: U x R” + R” x U’ by ~(x, t, J) = (y, hx, t). We then claim that q,i,*I(KxK’,
f,,Q)=j,*Z(KxK’,
rl,f2).
This will show that H, Z(K, @, U) = I(K’, Y, U’) preceding equalities imply that e@H,I(K,@,
since (*)
(*I and
the
U)=9*i,*I(KxK’,A,a)=j,*Z(KxK,,A,n) =e@Z(K),
Y, U’).
In order to prove (*) recall that for (x, y, t) ER we have T,(x,y, t) = (x, y) if and only if (x, y, t) E I7 := {(x, hx, t) 1(x, t) E Pj. Denote by 1: ZI-+ ?Z the inclusion. According to the definition of the index there is a JE H, I7 such that I(KxK’,f,,Q)=z,J. So it suffices to prove that q.+i,*z*=jl*l*. But j,: LJ -+ 58”~ u’ is homotopic to the map k: Q-r Wx u’ defined by k(x, y, t) := (hx, y, f). Since for (x, hx, t) E Z7 we have that vi, I(X, hx, t) = r/(x, hx, t) = (hx, hx, f) = kl(x, hx, t), we obtain q*i,*l,=k*l,=,j,*z,. We summarize PROPOSITION 1. For a simplicial complex K, a seml$‘ow ~0 on K, and an open set U c K x [0, “o) such that P := {(x, t) E U Ic$,.x = x) is compact there is an index I( K, @, U) E H, U.
Proof: We only have to prove the normalization property. Choose X~E C and a section S at x0 (this works in arbitrary metric spaces) with associated functions r and q. Then modify @ as in the corresponding proof for the euclidean case, embed K into some R” and argue as before. Q.E.D.
526
CHRISTIAN
C. FENSKE
Again, the index is defined (and homotopy-invariant and additive) if @ is just a continuous mapping @: K x [0, m) + K, and we shall use the same notation in this case without further notice. We now turn to the case where K is an infinite simplicial complex (with the Whitehead topology), UC K x [0, x) is open, cl D(U) is a compact subset of K, and P = {(x, t) E U 1cj, x - .x } IS compact. Then there is a finite subcomplex L of K such that cl Q(U) c L. Let U, = {(x, t) E U (x E L), denote the inclusion U,* + L by i and define I(K, @, U) := i, I( L, @, U,). Lemma 1 then immediately shows that I(K, @, U) does not depend on the particular subcomplex. Moreover, the index again satisfies properties (A), (H), and (N). As before, I( K, @, U) will be defined if we omit the assumption that @ be a semiflow. Generalizing further, we now assume that X is an ANR. If r is an open covering of X we denote by N, the nerve of a and by p1 any canonical pron U,, # Iz, we denote by jection pl: X-+ N,. If U,, .... U,, E M and U, n
We are going to show that for 6 tine enough the index I,< does neither depend on the particular pp, i, nor on the refinement /I of 6. If we choose 6 fine enough and fl finer than 6 then any two realizations i,], i;{: N,, + X of mesh 6 will be homotopic via a homotopy which is x-small. Since any two canonical projections X--t N,{ are homotopic we may invoke (H) to
INDEX
OF PERIODIC
527
ORBITS
conclude that Z,(X, @, U) does not depend on the particular pg, i,{ if we choose /I fine enough and the realization small enough. It remains to be shown that the index does neither depend on b if /I is finer than a certain 6. Choose an open covering 6 such that Z,JX, @, U) is defined if /3 is a refinement of 6. Since any two open coverings c1and /j’ have a common refinement it suflices to consider the case that we have a further refinement c( of /I. For UE c( then choose a p(U) E /3 with U c p(U) and generate a simplicial mapping qxa : N, -+ N, by qa8( ( U) .) = (,u( U) )fj. Now observe that the covering au /I is a refinement of /I, and a relines Mu b. Obviously, N, is a subcomplex of Nzvg. We define a retraction p: N,,, -+ N, as the simplicial mapping which maps (U), to qzB((U),) and ( V)/r to ( V)B. We claim that N, is in fact a deformation retract of N avg. Let then XEN,,~. The claim will be proven if we can show that (1 -i)x+lp(x)~N,,~ for i E [O, 11. Call G the minimal simplex containing x and denote the vertices of (Tby ( U, ), , .... ( U, ) 1, ( V, )a) .... ( V,)Lf. Then p(x) is contained in the simplex with vertices (p(U,)),, .... (P(U,))~, ( V1)B, .... (I’,),. But U, n ... n Ukn V, n ... n I’,#@ and U,cp(U,) for mc{l,...,k}, hence p(U,)n ... np(Uk)nU,n ... n Uk n P’, n ... n V,# @. This means that the simplex T with vertices < Ul>,, .*‘>(W., qrpWIM~ .... qx8KUkM~ (WB3 .... WJa belongs to N rvp. Since both x and p(x) belong to r, the straight line segment connecting x and p(x) is contained in r. We now denote the inclusion by z,~and compute N,+N,u,f
ZtN,,,, qzuscDj~~~~~j;~,Au)) = Z(N,,,, P4zus~~~u~~L~a(U)) (by (HII = z(N,,,, P~~~~~~,~~~~~&W) since both pII and pqZua are canonical projections = (Q x id),4Np,
Pp@j’jlvb)Cd, (U)n(N,,x
= (zfix id),4N,,
pII@j)J,,I13 .ig ‘(Wh
CO,~1)
by Lemma 1
hence I, v ,J X, @, U) = Z&X, @, U) since we may use i, u ,]zpas a realization of NB. On the other hand, call I, the inclusion N, -+ N,,,. Then z,p, is a canonical projection for the covering c1u/j’, so we obtain
z(N,,,, 41us@LsX&W =CL,, v,@Ll,J&W) = (I, x id),z(N,, p,@jg,j,,j; ‘(WI
according to Lemma 1.
528
CHRISTIAN C. FENSKE
So we have I, v ,(X, @, U) = Z,(X, @, U) which means that for sufficiently fine open coverings j3 the index ZB(X, @, U) does not depend on /I and we denote the common value by Z(X, @, U). So the index is well defined and we obtain PROPOSITION 2. For an ANR X, an open set U c Xx [0, LX)), and a semiflow @ on X such that cl Q(U) is compact and P = {(x, t) E cl U( 4,x = x) is a compact subset qf U there is an index Z(X, @, U) E H, U.
Proof In order to show additivity we use the notation of (A). Since P,, P, are compact subsets of U,, U, there are open sets V, , V, such that P,cV,cU,, PZCV/rCUZ, d(c1 V,, all,) > 0, d(c1 Vz, dUz) > 0. We may then choose the open covering a so fine that Z(X, @, U) = Z,(X, @, U) and ilfj,pcrx # x for all (x, t) 6 i,~‘( U)\i;‘(cl V, u cl V,). Call
the inclusions. We then have
z,(X @,u) =j,*W,,
p,%, j;‘(U))
=j,*Cklsz(N,7 Pa?i,,.jx-‘(V,)) +k2*4N,, p,@j’j,,L1(V2))1
by (A)
=jlfC~l.~I.z(N,, p,@j,Ji’(V,))
+ 1~,&,4N~,p,@jz,L1(Vd)l =ja*CzI.4N,, p’l@jl,.irl(Uil)) + 12.4N,, p,@jx, L’(UJ)l =ilJ,*z(N,, p,@jpj,,,j;m’(ul))
by (2)
+ iz,j,rz(N,, p,@jE,.i;‘(ud) = i,*Z,(X, @, U,)f i**I,(X @, U,). Since (H) is obvious it remains for us to show the normalization property. So we assume that P=Cx {p}, where C= {#,x,IOdt
INDEX OF PERIODIC ORBITS
529
now is the following: By the definition we have to choose an open covering 6 of X and to compute Z(X, @, v, x (p - 2E,p + 2s)) = Z,(X, @, vo x (p- 2E,p + 2s)) and ind(X, F( ., p), 0) = ind(N,, p,F( ., p) i,, i;‘(O)). Now we could apply the result for simplicial complexes if we could show ~~qS~i~x=x} is a one-sphere that P,= {(x, t)~ib’(V~)x(p-2~,p+2~)( and that p,F( ., p) i6 has precisely one fixed point in i; ‘(0). Of course, one cannot expect this to happen. Note, however, that (by (1)) in computing Z(N,, ps@j6, iy’( V,) x (p--2&, p + 2s)) we may replace P, by a larger compact subset K of i; ‘( V,) x (p - 2s, p + 2s). By a judicious choice of the open covering we will find a set K which has the homotopy-type of a onesphere. In the same way, one shows that one can find a contractible subset of is-‘(e) containing the fixed point set of p,F(., p) i,. We carry out the argument for the semiflow; the (simpler) argument for F( ., p) will then be clear. We choose an open set Vi, with Cc P’bc cl V0c V,, and an open covering cq,of X such that for each open refinement c( of a, we have that 1(X,@,
V,x(p-2&,P+2&))=j,*Z(N,,P,~i,,i,’(~,)x(P-2&,P+2&))
provided i, has mesh cr,; - Z,(X, @, V0 x (p - 28, p + 2.5)) does neither depend on p1 nor on i,; - x # pz#,i,x if x E V0 and E< It - pi < 2s provided i, has mesh Q. We now choose an open refinement c1 of G(~such that each partial realization of a simplicial complex with mesh CIextends to a full realization with mesh cq,. Choose an open covering { Ur, .... U,} of C such that the nerve N’ of (U,, ,.., U,} is a (simplicial) one-sphere (i.e., each Ui meets precisely two other sets of the covering) and such that each U, is contained in a member of ~1. Now we choose an open covering /I of X/C with the following properties: -if x~cl Vb\C is contained in VEX and If--pi ~2s then Vn 4,(V) = 0; - each element of p is contained in an element of CL Then we choose an open locally finite refinement y of /? (covering X\C) such that N, has a realization i,: N, -+ X\C which is p-small. (Note that X\C is an open subset of an ANR, hence an ANR itself.) Call 6 the locally finite open covering consisting of y and {U,, .... U,,). We have a partial
530
CHRISTIAN C. FENSKE
realization i of N6 with iI N, = i, such that i maps N’ homeomorphically onto C. We extend i over N, and obtain a full realization i,: N, + X with meshcc.IfxEi;-‘(V,)nN,andIt-pl<2swedenoteby (V,),,...,(V,), the vertices of the minimal simplex containing x. Then there is a set WE /II containing Vi, .... I/, and i7( ( V, , .... V,);.). But W n q5,(W) = @, so hence $riGx$ fly= 1 Vi which implies that pvq5riyx$ (V, .... Vm)y, This means, however, that Pa=((x,t)Eib’(Vo)x pyq5,i7x #x. x } is contained in the open star of N’. Since for a (P-~E,~+~E)IP~~~~~X= vertex ( V)& of IV6 which is adjacent to N’ and (t - pl < 2s we have that (( V),, t) $ P6 we seethat there is in fact a closed set K containing P6 such that N’ is a strong deformation retract of K. Since 6 is locally finite, K must be compact. In the same way, we see that we may choose 6 in such a way that {xE i;‘(O)1 paF(isx, p) = x } IS contained in a compact contractible set B (for example, a compact contractible subset of the open star of p,x, in N,). Applying the corresponding result for simplicial complexes and observing that K is homotopy equivalent to a one-sphere we obtain = ind(N,, p,F( ., p) i,, ii’(c!)).
[ps(C)]
hence Z(X, @, V, x (p - 25 p + 2~)) = i(@“) . [Cl. Before we turn to the final step we need a preparatory result LEMMA 2. Let X be an ANR, U c Xx [0, CG) open and @Ja semiflow on X such that cl Q(U) is compact and P = ((x, t) E cl UI I$,X = x} is a compact subset of U. Let Yc X be open and assume that cl 4(U) c Y. Let U,= {(x, t) E UI x E Y} and denote by i: U, + U the inclusion. Then Z(X, @, U) = i,Z( Y, @, U,).
Proof. Since cl @j(U)c Y we see that the closed set P is contained in U,, so the index on the right-hand side is defined. Since the compact set
cl @p(U)is at positive distance from X\ Y we may choose open coverings CI of X and a’ of Y such that -for each open refinement b of c( and fl’ of CI’ and realization i,: N, + X, i,. : N,, -+ Y which is x-small (resp. cr’-small) we have that m @, U) = Z/3(X @, U), I( K @, U,) = I,4 y, @, U,), -if VEC~UC~’then (VEa’ and cl@(U)nst,(V)#fa) if and only if (VEX’ and cl Q(U) n st,.( V) # @), where st, is the open star of V with respect to a. We then choose locally finite open refinements b of a and fl’ of x’ such that - {V~plcl@(U)n I’#@} and {VEIJ’Icl@(U)n V#@} are finite. -if VE/?U~’ then (V~fl and cl@(U)nsts(V)#QJ) if and only if (V~p’and cl@(U)nst,,(V)#12/).
INDEX OF PERIODIC ORBITS
531
We call L the finite subcomplex of N, (and of NP,) consisting of all simplices ( V,, .... P’,) with at least one V, intersecting cl Q(U). Then we choose realizations ip: N, -+ X, i,, : NB,+ Y such that ip(ip,) is a-small Let j,‘(U),= {(x, t)~Xx [0, co)1 (a’-small) and i, I L = i,. I L. (iax, t) E U, x E L}. By our choice of refinements and realizations we have that j,‘(U),cj~.~(U,). Call Z:j;l(U)r-+j~‘(Ur) and k: j,‘(U),)-+ j,‘(U) the inclusions. The rja,l = j,k. So we have that 1(X, @, U)=I,dX,
@, U)=j~*Wf~,
=jp*k*4L,
pgQjjg,j~Yu))
pp@jp, j,‘(u)d
by the definition of the index for infinite complexes =j&Z(L,
ppz@jas,jp”l(U)L)
= i, js.*l,Z(L,
pg,@jij,,, j,‘(U),)
=i*j8,*Z(NgS,
psf@jbSj,., j;‘(U,))
= i,ZDJ Y, @, U,) = i,Z( Y, 0, U,)
which proves our claim. COROLLARY. Let X, @, U, P be as in Lemma 2. Assume that Y c X is open and that PC U,. Then Z(X, @, U) = i.+( Y, @, U,), where i: U, -+ U is the inclusion.
Proof: As in (1) we see that Z(X, @, U) = i,Z(X, @, U,). Then we apply Q.E.D. Lemma 2 to conclude that Z(X, @, U,) = Z( Y, @, U,).
Finally we turn to the situation described in the beginning of Section 2. THEOREM.
Let X be an ANR and CDa semiflow on X such that
-
there is a to > 0 such that q4,,is locally compact and
-
@ has a compact attractor.
Let UC Xx [0, 00) be open and call P := {(x, t)~cl Ul#tx=x}. Assume that pr,p is a bounded subset of (t,, co) and that P c U. Then the index Z(X, @, U) is defined and satisfies (A), (H), and (N). Proof: Choose an open set V with PC Vc U such that pr, V is a bounded subset of (I,, co). According to [ 11 there is a compact attractor A for Qi which has arbitrarily small neighborhoods Y such that @(Y x [0, co)) c Y and cl d,,,Y is a compact subset of Y. Choose such a set Y and call V, := {(x, t) E VI x E Y}. Then define Z(X, @, U) := i, Z( Y, @, Vr), where i: V, + U is the inclusion. The index on the right-hand side is defined since cl @(Vy) is a compact subset of Y. Moreover, it is
532
CHRISTIAN C. FENSKE
obvious (by Proposition 2) that the index satisfies (A), (H), and (N). So the only thing we have to verify is that the index does neither depend on the choice of V nor of Y. Since the independence of V is immediate (apply (2)) we shall assume that pr2 U is already a bounded subset of (to, co). Let then Z be an open set such that @(Zx [0, co))cZ, cl d,,Z is a compact subset of Z, and x E Z whenever (x, t) E P. Since Zn Y will satisfy the same requirements it suffices to consider the case Zc Y. Denote by k the inclusion U, + U,. Then we have to show that i,k, I (Z, @, U,)=i,Z(Y, @, U,). But k,l(Z, @, U,)=Z(Y, CD,U,) by the corollary. 4. CONCLUDING REMARKS We now have seen that there is a well-defined index Z(X, @, U) satisfying the additivity, homotopy, and normalization property in an infinite dimensional situation without any differentiability assumptions. There is, however, a draw-back noticed already by Fuller [2]: If C is a single attracting periodic orbit we might have 1(X, Q’, U) = 0 if we choose U so large that [C] bounds in U. Fuller’s main achievement in [2] was to construct a normalized index (a rational number rather than an integral homology class) avoiding this difficulty. On the other hand, it is obvious that a single periodic orbit C always has small neighborhoods U such that [C] represents a nonzero element of H, U. More formally, one could define Z(X, @, U) = (Z(X, @, V)), where I/ ranges over the directed system of open c V. This index would also satisfy sets Vc U with P= {(x, t)~ Uld,x=x} properties (A), (H), and (N) if properly formulated.
REFERENCES 1. C. C. FENSKE, Periodic orbits of semiflows-Local indices and sections, in “Selected Topics in Operations Research and Mathematical Economics” (Lecture Notes in Economics and Mathematical Systems Vol. 226) pp. 348-360, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1984. 2. F. B. FULLER, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967) 133-148. 3. S.-T. Hu, “Theory of Retracts,” Wayne State Univ. Press, Detroit, 1965. 4. M. A. KRASNOSEL’SKI~ AND P. P. ZABREIKO, “Geometric methods of nonlinear analysis,” Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1984. 5. W. S. MASSEY, “Homology and Cohomology Theory,” Dekker, New York/Base], 1978.
OF MATHEMATICAL
ANALYSIS
AND
APPLICATIONS
129, 517-532 (1988)
A Simple-Minded Approach to the Index of Periodic Orbits CHRISTIAN
C. FENSKE
Mathematisches Institut, Justus-Liehig-Univer.vitiit. Arndtstrasse 2, Dd300 Giessen, West German? Submitted by Ky Fan
Recevied July 17, 1986
We define an index for periodic orbits of semiflows which satisfy a mild compactness condition on infinite dimensional ANR spaces without requiring differentiability. The index is a one-dimensional homology class. I( 19XX Academ,c Presr, Inc
1. INTRODUCTION In [2], F. B. Fuller defined an index for periodic solutions of autonomous differential equations on manifolds. In fact, his article describes several indices and we shall concentrate here on the approach which defines the index of a periodic solution as a one-dimensional integral homology class. Fuller’s approach used cohomology of differential forms, so he had natural assumptions on differentiability whereas we intend to deal with the continuous case. So we do not consider periodic solutions to a differential equation but rather periodic orbits of a flow-or, more generally, a semiflow since we shall be mostly interested in an infinite dimensional situation. A semiflow on a topological space X is a continuous mapping @: Xx [0, co) -+ X such that c$,,= id and dffs = d,#, for s, t 2 0 where we have written ~,4,=@(., t) for 220. Semiflows naturally occur in the study of retarded differential equations and it is there that one is interested in getting rid of differentiability assumptions on the semiflow. As to the underlying phase space X we shall assume that X is an ANR and we refer the reader to [3] for material concerning ANR spaces.We will start by considering a situation where X is open in a euclidean space, then we generalize to simplical complexes, and finally we approximate an ANR by nerves of open coverings. The index we shall define will be additive, homotopy invariant, and in case of a single periodic orbit the index can be described in terms of (the classical fixed point index of) the Poincart map (which requires some extra proviso in the case of semiflows). 517 0022-247X/88 $3.00 CopyrIght m(, I988 by Academic Press, Inc All rights of reproductmn I” any form reserved
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CHRISTIAN
C. FENSKE
2. THE SETUP Let X be an ANR and @ a semiflow on X such that - there is a to B 0 such that $,, is locally compact (i.e., each x E X has a neighborhood I/ such that cl d,,(V) is compact) - @ has a compact attractor (i.e., there is a compact set A such that for each neighborhood W of A and each x E X there is a t with dtx E W). Let UcXx [0, 00) be open and call P:= {(x, t)Ecl Ul$,x=x}. Assume that pr,(P) is a bounded subset of (to, co) (where pr, is, of course, the projection onto the second factor) and that P c U. Then P is in fact compact: according to [ 1] there is a compact attractor A for @ which has arbitrarily small neighborhoods V such that @(I’ x [0, co)) c I/ and cl q4,,(V) is a compact subset of V. It is obvious that all periodic points of @ must be contained in A, so P is compact. We shall say that there is defined an index of CDwith respect to U if there is an element Z(X, @, U) E Zf, U (the first homology group of U) such that (A) the index is additive: if P = P, u P,, P, c U1, P, c U, are closed, U,, U, are disjoint open sets with U, u U2 c U then 4X, @, U) = ileZ(X, @, U,) + &,Z(X 0, UJ, where i, : U, + U, i,: U2 + U are inclusions. (H) the index is homotopy-invariant: Assume that @:Xx[0,co)x[0,1]-+X x, t, s -+ @qx, t) is continuous and that each @” is a semiflow. Define Y: (Xx [0, 11) x [0, co) + Xx [0, l] by Y(x, S, t) = (@“(x, t), S) and assumethat Y is locally compact for some to 2 0 and has a compact attractor. Let UC Xx [0, co)x [0, l] be open and call P= {(x, t,S)Ecl U(@“(x, t)=x}. Assume that P c U and that prz(P) is a bounded subset of (to, co). Then i,*Z(X Go, U,) = i,.W,
@‘, U,),
where
Uo=(Xx[O,co)x{O})nU, U,=(Xx[O,co)x(l})nU, and i,: Uo+ U, i,: U, + U are inclusions. (N) the index is normalized: If P consists of a single periodic orbit C
then Z(X, @, U) = i(@“)[C],
INDEX
OF PERIODIC
519
ORBITS
where i(@‘) is the index of the Poincart map associated with C and [C] is the class of the singular l-cycle represented by C in U. (To be precise, let T be the minimal period of the orbit corresponding to C, then [C] is represented by [0, l] + U, t -+ @,,x for some x on C. For the definition of i(Qp”) we refer to [ 1 or 43, see also the proof of (N) in Section 3 below for a discussion of the relevant facts.)
3. THE
DEFINITION
The definition will be given in several steps. We start with the following situation: D = X is open in R”, U is open in D x [0, co), @:U -+ D is continuous, and P= {(x, t) E UI 4,~ =x} is a compact set. Call t the orientation r E H"(D x D, D x D\A), define g: U -+ D x D by g(x, t) = (x, 4,x), and define Z(D, @, U) to be the image of t under
p(DxD,DxD\A)
,,R"(U,U\P)zHfP~H,U,
where R* denotes Alexander-Spanier cohomology, z is Poincare duality [S, p. 3631, and i: P -+ U is the conclusion. We remark: (1) In the situation described above, let P c Q c U, Q compact. Then we may replace P by Q in the definition of Z(D, @, U):
JP(DxD,DxD\A)-+R”(U, u\Q)rHfQ-H,u
‘\,I
I/
A"(U, U\P)zH;P
where the vertical arrows are induced by inclusions (since Poincare duality is natural with respect to inclusions). (2) If V is an open subset of U sucht that P is a compact subset of V then Z(D, @, U) = j,Z(D, @, V), where j: V + U is the inclusion. This is obvious since i, : H; P -+ H, U factors through H, V. Property (A) is obvious if U= U, u U,. For the general case, use (2) with V= UI u Uz. As to (H), we prove this without using the assumption that the homotopy occurs through semiflows. For s E [0, l] let P, = {(x, t, s) E U 1@‘(x, t) = x} and choose a 6, > 0 and a compact set C, c X such that P,cC,x {t} c U as long as Is- tI d 6,. Choose an open set If7c Xx [0, co) such that
P,x([s-6,,s+6,]n
[O, l])c
lfyx [s-d5,s+d,]c
U.
520
CHRISTIAN
C. FENSKE
Denote by j, the inclusion j-V: V, x ([s - 6,, s + S,] n [0, 11) + U. We then have
io,Z(X,CD',Uo)=io,joeZ(X,@O,Vo)=ilJl*Z(X, @', Vl)=il*(X @', Ul) since by (1) we may replace P, by C, in the definition of Z(X, @j’,U) as long as Is - t) d 6,Y.We then choose a finite subcovering of [0, 1) and proceed by induction, for it is obvious that t --f Z(X, @‘, V.,x [s - 6,, s + d,]), does not depend on t as long as (s - tl < 6,. We now turn to the normalization property: We use the notation adopted in (N). We may choose U so small that P is the only periodic orbit in cl U and such that U is homeomorphic to C x B, where B is a small open ball in Iw”. There is a unique p > 0 such that P = { (dtxo, p) 10 < t < p} for some x0 E C. Choose a section S for C. This means that x0 E S and that there is a neighborhood V, of C in U, := Un ([w” x {p}) such that for each xE:C:=Sn v, the map defined by r(x):=inf(t>Ol4,x~S} is continuous, that 4,~ E U, whenever )’ E C and 0 d t < r(v), and that for each y E V, the number q(y) = inf( t > 0 14,y E S} is defined and q 1V\Z is continuous (cf. [ 11 for details). Moreover, we may (and shall) choose V. so small that jr(x) - r(x,)l < r(x,)/8 for XE V,. If @ were a flow it would be easy to show that Z is an ANR, but though a semiflow on D (or, more generally, a finite dimensional manifold) cannot have startpoints it does not necessarily have backward unicity, so the obvious neighborhood retraction onto Z along flowlines does not work. We therefore use a substitute for the Poincart map which will be defined in a neighborhood of C. Geometrically, think of @ as flowing from the left into C and leaving it to the right, i.e., define V, = (x E V, 1q(x) < z(x,)/4}, V, = {x E V, I q(x) > 3~(x,)/4}. We now speed up @ in V, a little bit and slow it down on V,. This will have the effect that x0 is the only fixed point of the modified @ in V,u V, and we use its index as the index of the Poincart map. We use a homotopy which does not affect @ on t3U to deform @ in a small neighborhood of x0 in the same way. Note that during all deformations no point leaves its flowline so that fixed or periodic points can occur only on C (but possibly for other values of the time parameter). The details are as follows: Call 0 := int(c1 V,u cl V,). Choose an E> 0 such that U, x (p - 28, p + 2) c U and define for x E cl V, if r(x,)/8 d V(X) $ $x,)/4 /l(x) := 7(-xo)/4-ul(x) if q(x) G 7(x,)/8 v(x) i and for x E cl V, if xES if 37(x0)/4 < q(x) d %(x,)/8 if 7~(x,)/8
INDEX
OF PERIODIC
Then extend ,U continuously over {x~cl V,I q(x)> 7$x,)/8} cl(X) E C-~(X,Y8,01. ff xEc1 6 and It-pi 622~ let
;I(x, t) := o(E- It - PI 1P(X)/&
521
ORBITS
such that
if It-Z7
Now we define a deformation h: cl V, x [p - 2c, p + 2E] x [0, l] + D connecting h(x, t, 0) = dtx with F(x, t) := h(x, t, 1) by h(x, t, ct) := 411(.Y,r)+rx i 4 ,X
if Odd(x,C)<2E and It-pi 62~ if d(x, E) 3 2~ or d(x, L’) < 2~ and c6 It-p1
d2E.
By definition, the in&x of the PoincarP map, i(@“), then is i(@“) := ind(D, F( ., p), 0) where ind is the classical fixed point index. Note that by (H) we have that Z(D,@,
v,X(p-2E,p+2E))=Z(D,F,
V,X(p-2E,p+2E)).
Now embed V,, into V via i(x) := (x, p) and call Z:= {(x, t)~ VI F(x, t) =x}. Then Z is a compact subset of V (and in fact homeomorphic to a one-sphere). Choose the obvious generator y E H, Zz H, C and call c the generator of I?( V, V\f) which is mapped onto y under the isomorphism Z?‘( V, V\Z) r H, ZY Write Z(D, @, V) = Z(D, F, V) = I. y and denote by t the orientation r E R”(D x D, D x D\d). So r is mapped onto I. c by the homomorphism l?(D x D, D x D\d) + R”( V, V\Z) which is induced by (x, t) -+ (x, F(x, t)). Now if n 3 3 we have that (with j: 0 + V being the inclusion)
H”(v, v\q
.I* * H”(O,~\{x,})
II I’H”-‘(~\{x”}) I ijn-‘(V\r) and all groups are isomorphic to ;2. Co\{x0} is homotopy equivalent to S”- ’ and V\Z is homotopy equivalent to S’ x S”-’ and it is easy to see that j* maps the generator of
onto the generator of H” ~ ‘(S\{ x,}) 2 H” ~ IS”- ‘. (A similar argument works if IZ= 2. We then have H2( V\Z) E Z @Z and an isomorphism H2( V, V\T) E H2(0, O\{x,)). But this means that the index of the Poin409,129’2-I2
522
CHRISTIAN
C. FENSKE
care map, which by definition is just the fixed point index of F( ., p), equals Z since the fixed point index of F( ., p) = Fo i is the image of r under R”(DxD,DxD\A)------+
WY
v\o
i*
fz”(G,
c?\{X”}).
This completes the proof of (N). At first glance it might seem that one could prove the normalization property (N) just by noting that Z(D, @, V) = J. [C] where J is the fixed point index of the map V+ D x [0, cc) given by (x, t) + (4,x, p). This does, however, not work-simply because it is false. The point is that we obviously have J=O; just reparametrize @ by a homotopy so that the periodic orbit occurs at a time parameter different from p. In addition, we note that instead of [C] we might have taken the class of t + d,pxO (whereas above we have used the class of t + 4,,x,). This would have to be compensated for by dividing i(@“) by p/r. We now turn to the case where X= K is a (finite) simplicial complex, @ is a semiflow on K, U is open in Kx [0, cc), and P= {(x, t)E Ulcj,x=x} is compact. We choose a PL-embedding h: K -+ R” into some R”, an open set D=J K, and a retraction r: D +/z(K). Let R :=r x id, H :=h x id, and define g: R-‘(H( U)) -+ D x D by g(x, t) := (x, h#,h-‘rx). We then define I,,, .(K, @, U) to be the image of the orientation TE R”(D x D, ~;~ZI~/)_~;der ~!“(DxD,DxD\~)--@*R~(R-~(U),R-‘(U)\H(P))E c * H, U. We are then left with the tedious job of showing that Zh,lr(K, @, U) does neither depend on the embedding nor on the retraction so that we may put Z(K, @, b’) := I,,, ,(K, @, U). We start by showing that the index does not depend on the retraction. In order to simplify our notation let us assume that K is already a subset of R”. First, it is obvious that we may replace D by a smaller neighborhood D’ of K and r by r 1D’. Let then rl: D, -+ K, r2: D, + K be retractions. By (2) we may replace D,, D, by D := D, n D,. Define q: (Dx {O})u (Kx[O, l])u(Dx {l})-+K by q(x,O)=r,x, q(x, t)=x, q(x, l)=rzx. Then we find an open neighborhood D’ of K and an extension Q of q over D’ x [IO, 1). Again, we may replace D by D’. So we are reduced to the situation where we have homotopic retractions r, , r2 : D -+ K and (H) shows independence. Now assume that K is PL-embedded into R”, that m > n, that D c iw”’ is a neighborhood of K x (0) in R”, and r: D + K is a retraction. As we have just seen we may assume that D = D, x W where D, is open in R”, W is a neighborhood of 0 in R”~“, and r = (r,p, 0), where p: R” + R” is the orthogonal projection and ro: Do + K is a retraction. Write points in R”’ as (x, y), XE R”, YE lR”-“. Then Z,(K, @, U) is by definition the image of the orientation
INDEX
OF PERIODIC
523
ORBITS
z,=T,,xT,ERm((DoxD,,DoXD,\A)x(Wx zA”(D,xD,,D,xD,\A)@~“-“(Wx
w, wx W\A)) W, Wx W\A)
which is mapped into B”(R-‘(U),
R-‘(U)\P)rET”(R,‘(U),
R,‘(U)\P)@P+“(W,
W\(O))
zHf(P)@Hg({O))EH;(P) where, of course, R = r x id, R, = r,, x id. This shows that Z,(K, @, U) = Z,W, @, U).
Before we show independence of the embedding we prove a weak form of the “commutativity property.” LEMMA 1. Let K be a simplicial complex in [w”, L a subcomplex of K. Let @ be a semifrow on K, UC K x [0, 00) open and assume that is compact and that C$,XEL whenever (x, t) E U. P= {(x, t)E Ul&x=x} Denote by i: {(x, t) E ZJ x E L} + U the inclusion. Then Z(K, @, U) = i,Z(L, @, i-‘(U)).
Prooj Choose a neighborhood D of K in R” and a retraction r: D + K. Let R = r x id. Let dp be a neighborhood of L in K such that there is a deformation retraction q: B + L and let Q = q x id. Then r’(9) is open in R” and we may use qr: r ~ ‘(9) + L as a retraction for L. Then Z(L, @, i- l(U)) may be defined using the map @iQR: R-‘(-Y x [0, co))n Q-‘(Un (Lx [0, co)) -+ L. But iQ is homotopic to the identity, so (H) Q.E.D. shows that the index of @iQR equals the index of @R.
Finally, we show independence of the embedding. In order to simplify our notation we assume that K is a simplical complex in IR” and that h: K + K’ is a PL-homeomorphism. By what we have seen above we may assume that K and K’ are subsets of the same R”. Let U be as above, H:=hxid, U’=H(U).Deline Y:Rx[O,oo)+K’by$,y:=hd,h-‘y.We then have to show that Z(K’, Y, U’)= H,Z(K, @, U).
This is achieved by mimicking the proof of the commutativity property for the classical fixed point index. We start by replacing U by a smaller open set so that we may assume dlx # x whenever (x, t) E au. Choose open sets D, D’ in R” and retractions r: D -+ K, r’: D’ + K’. Define a ((2n + 1)-dimensional) open set Q := ((4 y, t)I (rx, t) E U and (r’y, t) E U’ )
524
CHRISTIAN
C. FENSKE
and a map by /l(x, I’, t) = (4,h- ‘r’!:, hrx).
A:KxK’x[O,x~)+KxK
We then define a homotopy l? .Q x [0, l] + K x R” by fJx, J‘, t) := (d,h ‘r’y, xhd,h ‘Y’Y + (1 - c() hrx). We claim that Z-Jx, ~1,t) # (x, y) if (x, y, t) E 8Q and LXE [0, 11. Indeed, assume that T,(x, I’, t) = (x, ~1) then ~EK hence rx=.x and x=cj,h ‘r’y, so hx=zhx+(l-r)hx=.t’. So we have that y E K’ hence r’y = y. This yields #,x=x, consequently (.v, t) $ dZJ, but then also (y, t) 4 au’ since 4’ = hx. So (x, y, t) 4 6X2. In particular, we have seen that f,(x, y, t)=(x, y)for (x, y, z)ESL ifand only if (x, ~)EP:= ((x, t)~ U/dIx=.x} and y=hx. In th e same way we see that T,(x, y, t) = (x, ~1)for x E IF!“, (y, t) E U’, implies (x, t) E U. Denote byj, : Q -+ IF!”x U’ the inclusion. Using (2) we then obtain ,j,*Z(Kx
K’, A,Q)=j,*l(KxK’,
f,,Q)=Z(R”xK’,
f,,
Choose a point x,, E K and define a homotopy R” x K’ by B,(x,
I’, t) := (( 1 -r)
LYx U’).
B: R” x U’ x [0, 1] -+
/,d,h ‘r’y + c~x~, h#,h
‘r’y).
We claim that (x, ~1)# B,(x, ,v, t) if (I’, t) E au’, C(E[0, 11. Assume that B,(x, I’, t) = (x, ~1) then YE K’, so r’y = J and hcj,hk’y = JJ which is impossible if (y, t) E au’. So I(IW”xK’,~,,[W”xU’)=Z([W”xK’,
B,,WxU’).
Now B, is a product of a constant map and Y, so we obtain I(R”xK’,
B,, Wx
U’)=e@I(K’,
Y, U’).
where e is the generator of H,,R”. On the other hand, consider the homotopy 0: .Q x [0, l] + R” x K which is given by 0,(x, y, t) = (coj,rx+ (1 -z) 4thk’r’y, hrx). Again, we claim that 0,(x, I’, t) # (x, y) for (x, y, ~)ELMX, E E [0, 11. Indeed, if 0,(x, I’, t) = (x, ~7) then y E K’ hence r’jl= J’ and hrx = y. So 4,h -‘y = xd,h-‘y+(l-z)4,/1 ‘~=~,rx=x.ButthenxEKandrx=.x,sowehave 4,x=x, hence (x, t) $ iiU, and hx = y, hence (J, t) 4 8U’. Denote by i, the inclusion i, : Q -+ CJx R”. As above, we have that
525
INDEX OF PERIODIC ORBITS
Finally, we consider the homotopy Z:UxKY’x[O,l]~KxR”whichis given by &J-x, t, 4’) = (d,vx, (1 - c() hrx + CCX~). It is immediate Z,(x, t, y) # (x, I’) if (x, t) E au, and we have that I(Kx
R’“, E,, Ux R”)=Z(K,
that
CD,V)@e.
Putting things together we see that j,*I(Kx
K’, A, Q)=e@Z(K’,
Y, u’)
and i,*I(KxK’,A,Q)=I(K,@,
U)@e.
Now define ‘1: U x R” + R” x U’ by ~(x, t, J) = (y, hx, t). We then claim that q,i,*I(KxK’,
f,,Q)=j,*Z(KxK’,
rl,f2).
This will show that H, Z(K, @, U) = I(K’, Y, U’) preceding equalities imply that e@H,I(K,@,
since (*)
(*I and
the
U)=9*i,*I(KxK’,A,a)=j,*Z(KxK,,A,n) =e@Z(K),
Y, U’).
In order to prove (*) recall that for (x, y, t) ER we have T,(x,y, t) = (x, y) if and only if (x, y, t) E I7 := {(x, hx, t) 1(x, t) E Pj. Denote by 1: ZI-+ ?Z the inclusion. According to the definition of the index there is a JE H, I7 such that I(KxK’,f,,Q)=z,J. So it suffices to prove that q.+i,*z*=jl*l*. But j,: LJ -+ 58”~ u’ is homotopic to the map k: Q-r Wx u’ defined by k(x, y, t) := (hx, y, f). Since for (x, hx, t) E Z7 we have that vi, I(X, hx, t) = r/(x, hx, t) = (hx, hx, f) = kl(x, hx, t), we obtain q*i,*l,=k*l,=,j,*z,. We summarize PROPOSITION 1. For a simplicial complex K, a seml$‘ow ~0 on K, and an open set U c K x [0, “o) such that P := {(x, t) E U Ic$,.x = x) is compact there is an index I( K, @, U) E H, U.
Proof: We only have to prove the normalization property. Choose X~E C and a section S at x0 (this works in arbitrary metric spaces) with associated functions r and q. Then modify @ as in the corresponding proof for the euclidean case, embed K into some R” and argue as before. Q.E.D.
526
CHRISTIAN
C. FENSKE
Again, the index is defined (and homotopy-invariant and additive) if @ is just a continuous mapping @: K x [0, m) + K, and we shall use the same notation in this case without further notice. We now turn to the case where K is an infinite simplicial complex (with the Whitehead topology), UC K x [0, x) is open, cl D(U) is a compact subset of K, and P = {(x, t) E U 1cj, x - .x } IS compact. Then there is a finite subcomplex L of K such that cl Q(U) c L. Let U, = {(x, t) E U (x E L), denote the inclusion U,* + L by i and define I(K, @, U) := i, I( L, @, U,). Lemma 1 then immediately shows that I(K, @, U) does not depend on the particular subcomplex. Moreover, the index again satisfies properties (A), (H), and (N). As before, I( K, @, U) will be defined if we omit the assumption that @ be a semiflow. Generalizing further, we now assume that X is an ANR. If r is an open covering of X we denote by N, the nerve of a and by p1 any canonical pron U,, # Iz, we denote by jection pl: X-+ N,. If U,, .... U,, E M and U, n
We are going to show that for 6 tine enough the index I,< does neither depend on the particular pp, i, nor on the refinement /I of 6. If we choose 6 fine enough and fl finer than 6 then any two realizations i,], i;{: N,, + X of mesh 6 will be homotopic via a homotopy which is x-small. Since any two canonical projections X--t N,{ are homotopic we may invoke (H) to
INDEX
OF PERIODIC
527
ORBITS
conclude that Z,(X, @, U) does not depend on the particular pg, i,{ if we choose /I fine enough and the realization small enough. It remains to be shown that the index does neither depend on b if /I is finer than a certain 6. Choose an open covering 6 such that Z,JX, @, U) is defined if /3 is a refinement of 6. Since any two open coverings c1and /j’ have a common refinement it suflices to consider the case that we have a further refinement c( of /I. For UE c( then choose a p(U) E /3 with U c p(U) and generate a simplicial mapping qxa : N, -+ N, by qa8( ( U) .) = (,u( U) )fj. Now observe that the covering au /I is a refinement of /I, and a relines Mu b. Obviously, N, is a subcomplex of Nzvg. We define a retraction p: N,,, -+ N, as the simplicial mapping which maps (U), to qzB((U),) and ( V)/r to ( V)B. We claim that N, is in fact a deformation retract of N avg. Let then XEN,,~. The claim will be proven if we can show that (1 -i)x+lp(x)~N,,~ for i E [O, 11. Call G the minimal simplex containing x and denote the vertices of (Tby ( U, ), , .... ( U, ) 1, ( V, )a) .... ( V,)Lf. Then p(x) is contained in the simplex with vertices (p(U,)),, .... (P(U,))~, ( V1)B, .... (I’,),. But U, n ... n Ukn V, n ... n I’,#@ and U,cp(U,) for mc{l,...,k}, hence p(U,)n ... np(Uk)nU,n ... n Uk n P’, n ... n V,# @. This means that the simplex T with vertices < Ul>,, .*‘>(W., qrpWIM~ .... qx8KUkM~ (WB3 .... WJa belongs to N rvp. Since both x and p(x) belong to r, the straight line segment connecting x and p(x) is contained in r. We now denote the inclusion by z,~and compute N,+N,u,f
ZtN,,,, qzuscDj~~~~~j;~,Au)) = Z(N,,,, P4zus~~~u~~L~a(U)) (by (HII = z(N,,,, P~~~~~~,~~~~~&W) since both pII and pqZua are canonical projections = (Q x id),4Np,
Pp@j’jlvb)Cd, (U)n(N,,x
= (zfix id),4N,,
pII@j)J,,I13 .ig ‘(Wh
CO,~1)
by Lemma 1
hence I, v ,J X, @, U) = Z&X, @, U) since we may use i, u ,]zpas a realization of NB. On the other hand, call I, the inclusion N, -+ N,,,. Then z,p, is a canonical projection for the covering c1u/j’, so we obtain
z(N,,,, 41us@LsX&W =CL,, v,@Ll,J&W) = (I, x id),z(N,, p,@jg,j,,j; ‘(WI
according to Lemma 1.
528
CHRISTIAN C. FENSKE
So we have I, v ,(X, @, U) = Z,(X, @, U) which means that for sufficiently fine open coverings j3 the index ZB(X, @, U) does not depend on /I and we denote the common value by Z(X, @, U). So the index is well defined and we obtain PROPOSITION 2. For an ANR X, an open set U c Xx [0, LX)), and a semiflow @ on X such that cl Q(U) is compact and P = {(x, t) E cl U( 4,x = x) is a compact subset qf U there is an index Z(X, @, U) E H, U.
Proof In order to show additivity we use the notation of (A). Since P,, P, are compact subsets of U,, U, there are open sets V, , V, such that P,cV,cU,, PZCV/rCUZ, d(c1 V,, all,) > 0, d(c1 Vz, dUz) > 0. We may then choose the open covering a so fine that Z(X, @, U) = Z,(X, @, U) and ilfj,pcrx # x for all (x, t) 6 i,~‘( U)\i;‘(cl V, u cl V,). Call
the inclusions. We then have
z,(X @,u) =j,*W,,
p,%, j;‘(U))
=j,*Cklsz(N,7 Pa?i,,.jx-‘(V,)) +k2*4N,, p,@j’j,,L1(V2))1
by (A)
=jlfC~l.~I.z(N,, p,@j,Ji’(V,))
+ 1~,&,4N~,p,@jz,L1(Vd)l =ja*CzI.4N,, p’l@jl,.irl(Uil)) + 12.4N,, p,@jx, L’(UJ)l =ilJ,*z(N,, p,@jpj,,,j;m’(ul))
by (2)
+ iz,j,rz(N,, p,@jE,.i;‘(ud) = i,*Z,(X, @, U,)f i**I,(X @, U,). Since (H) is obvious it remains for us to show the normalization property. So we assume that P=Cx {p}, where C= {#,x,IOdt
INDEX OF PERIODIC ORBITS
529
now is the following: By the definition we have to choose an open covering 6 of X and to compute Z(X, @, v, x (p - 2E,p + 2s)) = Z,(X, @, vo x (p- 2E,p + 2s)) and ind(X, F( ., p), 0) = ind(N,, p,F( ., p) i,, i;‘(O)). Now we could apply the result for simplicial complexes if we could show ~~qS~i~x=x} is a one-sphere that P,= {(x, t)~ib’(V~)x(p-2~,p+2~)( and that p,F( ., p) i6 has precisely one fixed point in i; ‘(0). Of course, one cannot expect this to happen. Note, however, that (by (1)) in computing Z(N,, ps@j6, iy’( V,) x (p--2&, p + 2s)) we may replace P, by a larger compact subset K of i; ‘( V,) x (p - 2s, p + 2s). By a judicious choice of the open covering we will find a set K which has the homotopy-type of a onesphere. In the same way, one shows that one can find a contractible subset of is-‘(e) containing the fixed point set of p,F(., p) i,. We carry out the argument for the semiflow; the (simpler) argument for F( ., p) will then be clear. We choose an open set Vi, with Cc P’bc cl V0c V,, and an open covering cq,of X such that for each open refinement c( of a, we have that 1(X,@,
V,x(p-2&,P+2&))=j,*Z(N,,P,~i,,i,’(~,)x(P-2&,P+2&))
provided i, has mesh cr,; - Z,(X, @, V0 x (p - 28, p + 2.5)) does neither depend on p1 nor on i,; - x # pz#,i,x if x E V0 and E< It - pi < 2s provided i, has mesh Q. We now choose an open refinement c1 of G(~such that each partial realization of a simplicial complex with mesh CIextends to a full realization with mesh cq,. Choose an open covering { Ur, .... U,} of C such that the nerve N’ of (U,, ,.., U,} is a (simplicial) one-sphere (i.e., each Ui meets precisely two other sets of the covering) and such that each U, is contained in a member of ~1. Now we choose an open covering /I of X/C with the following properties: -if x~cl Vb\C is contained in VEX and If--pi ~2s then Vn 4,(V) = 0; - each element of p is contained in an element of CL Then we choose an open locally finite refinement y of /? (covering X\C) such that N, has a realization i,: N, -+ X\C which is p-small. (Note that X\C is an open subset of an ANR, hence an ANR itself.) Call 6 the locally finite open covering consisting of y and {U,, .... U,,). We have a partial
530
CHRISTIAN C. FENSKE
realization i of N6 with iI N, = i, such that i maps N’ homeomorphically onto C. We extend i over N, and obtain a full realization i,: N, + X with meshcc.IfxEi;-‘(V,)nN,andIt-pl<2swedenoteby (V,),,...,(V,), the vertices of the minimal simplex containing x. Then there is a set WE /II containing Vi, .... I/, and i7( ( V, , .... V,);.). But W n q5,(W) = @, so hence $riGx$ fly= 1 Vi which implies that pvq5riyx$ (V, .... Vm)y, This means, however, that Pa=((x,t)Eib’(Vo)x pyq5,i7x #x. x } is contained in the open star of N’. Since for a (P-~E,~+~E)IP~~~~~X= vertex ( V)& of IV6 which is adjacent to N’ and (t - pl < 2s we have that (( V),, t) $ P6 we seethat there is in fact a closed set K containing P6 such that N’ is a strong deformation retract of K. Since 6 is locally finite, K must be compact. In the same way, we see that we may choose 6 in such a way that {xE i;‘(O)1 paF(isx, p) = x } IS contained in a compact contractible set B (for example, a compact contractible subset of the open star of p,x, in N,). Applying the corresponding result for simplicial complexes and observing that K is homotopy equivalent to a one-sphere we obtain = ind(N,, p,F( ., p) i,, ii’(c!)).
[ps(C)]
hence Z(X, @, V, x (p - 25 p + 2~)) = i(@“) . [Cl. Before we turn to the final step we need a preparatory result LEMMA 2. Let X be an ANR, U c Xx [0, CG) open and @Ja semiflow on X such that cl Q(U) is compact and P = ((x, t) E cl UI I$,X = x} is a compact subset of U. Let Yc X be open and assume that cl 4(U) c Y. Let U,= {(x, t) E UI x E Y} and denote by i: U, + U the inclusion. Then Z(X, @, U) = i,Z( Y, @, U,).
Proof. Since cl @j(U)c Y we see that the closed set P is contained in U,, so the index on the right-hand side is defined. Since the compact set
cl @p(U)is at positive distance from X\ Y we may choose open coverings CI of X and a’ of Y such that -for each open refinement b of c( and fl’ of CI’ and realization i,: N, + X, i,. : N,, -+ Y which is x-small (resp. cr’-small) we have that m @, U) = Z/3(X @, U), I( K @, U,) = I,4 y, @, U,), -if VEC~UC~’then (VEa’ and cl@(U)nst,(V)#fa) if and only if (VEX’ and cl Q(U) n st,.( V) # @), where st, is the open star of V with respect to a. We then choose locally finite open refinements b of a and fl’ of x’ such that - {V~plcl@(U)n I’#@} and {VEIJ’Icl@(U)n V#@} are finite. -if VE/?U~’ then (V~fl and cl@(U)nsts(V)#QJ) if and only if (V~p’and cl@(U)nst,,(V)#12/).
INDEX OF PERIODIC ORBITS
531
We call L the finite subcomplex of N, (and of NP,) consisting of all simplices ( V,, .... P’,) with at least one V, intersecting cl Q(U). Then we choose realizations ip: N, -+ X, i,, : NB,+ Y such that ip(ip,) is a-small Let j,‘(U),= {(x, t)~Xx [0, co)1 (a’-small) and i, I L = i,. I L. (iax, t) E U, x E L}. By our choice of refinements and realizations we have that j,‘(U),cj~.~(U,). Call Z:j;l(U)r-+j~‘(Ur) and k: j,‘(U),)-+ j,‘(U) the inclusions. The rja,l = j,k. So we have that 1(X, @, U)=I,dX,
@, U)=j~*Wf~,
=jp*k*4L,
pgQjjg,j~Yu))
pp@jp, j,‘(u)d
by the definition of the index for infinite complexes =j&Z(L,
ppz@jas,jp”l(U)L)
= i, js.*l,Z(L,
pg,@jij,,, j,‘(U),)
=i*j8,*Z(NgS,
psf@jbSj,., j;‘(U,))
= i,ZDJ Y, @, U,) = i,Z( Y, 0, U,)
which proves our claim. COROLLARY. Let X, @, U, P be as in Lemma 2. Assume that Y c X is open and that PC U,. Then Z(X, @, U) = i.+( Y, @, U,), where i: U, -+ U is the inclusion.
Proof: As in (1) we see that Z(X, @, U) = i,Z(X, @, U,). Then we apply Q.E.D. Lemma 2 to conclude that Z(X, @, U,) = Z( Y, @, U,).
Finally we turn to the situation described in the beginning of Section 2. THEOREM.
Let X be an ANR and CDa semiflow on X such that
-
there is a to > 0 such that q4,,is locally compact and
-
@ has a compact attractor.
Let UC Xx [0, 00) be open and call P := {(x, t)~cl Ul#tx=x}. Assume that pr,p is a bounded subset of (t,, co) and that P c U. Then the index Z(X, @, U) is defined and satisfies (A), (H), and (N). Proof: Choose an open set V with PC Vc U such that pr, V is a bounded subset of (I,, co). According to [ 11 there is a compact attractor A for Qi which has arbitrarily small neighborhoods Y such that @(Y x [0, co)) c Y and cl d,,,Y is a compact subset of Y. Choose such a set Y and call V, := {(x, t) E VI x E Y}. Then define Z(X, @, U) := i, Z( Y, @, Vr), where i: V, + U is the inclusion. The index on the right-hand side is defined since cl @(Vy) is a compact subset of Y. Moreover, it is
532
CHRISTIAN C. FENSKE
obvious (by Proposition 2) that the index satisfies (A), (H), and (N). So the only thing we have to verify is that the index does neither depend on the choice of V nor of Y. Since the independence of V is immediate (apply (2)) we shall assume that pr2 U is already a bounded subset of (to, co). Let then Z be an open set such that @(Zx [0, co))cZ, cl d,,Z is a compact subset of Z, and x E Z whenever (x, t) E P. Since Zn Y will satisfy the same requirements it suffices to consider the case Zc Y. Denote by k the inclusion U, + U,. Then we have to show that i,k, I (Z, @, U,)=i,Z(Y, @, U,). But k,l(Z, @, U,)=Z(Y, CD,U,) by the corollary. 4. CONCLUDING REMARKS We now have seen that there is a well-defined index Z(X, @, U) satisfying the additivity, homotopy, and normalization property in an infinite dimensional situation without any differentiability assumptions. There is, however, a draw-back noticed already by Fuller [2]: If C is a single attracting periodic orbit we might have 1(X, Q’, U) = 0 if we choose U so large that [C] bounds in U. Fuller’s main achievement in [2] was to construct a normalized index (a rational number rather than an integral homology class) avoiding this difficulty. On the other hand, it is obvious that a single periodic orbit C always has small neighborhoods U such that [C] represents a nonzero element of H, U. More formally, one could define Z(X, @, U) = (Z(X, @, V)), where I/ ranges over the directed system of open c V. This index would also satisfy sets Vc U with P= {(x, t)~ Uld,x=x} properties (A), (H), and (N) if properly formulated.
REFERENCES 1. C. C. FENSKE, Periodic orbits of semiflows-Local indices and sections, in “Selected Topics in Operations Research and Mathematical Economics” (Lecture Notes in Economics and Mathematical Systems Vol. 226) pp. 348-360, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1984. 2. F. B. FULLER, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967) 133-148. 3. S.-T. Hu, “Theory of Retracts,” Wayne State Univ. Press, Detroit, 1965. 4. M. A. KRASNOSEL’SKI~ AND P. P. ZABREIKO, “Geometric methods of nonlinear analysis,” Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1984. 5. W. S. MASSEY, “Homology and Cohomology Theory,” Dekker, New York/Base], 1978.