Isolating neighborhoods and chaosbyKonstantin Mischaikow1;2 and Marian Mrozek 3September, 1994Abstract. We show that the map part of the discrete Conley index carries infor-mation which can be used to detect the existence of connections in the repeller-attractordecomposition of an isolated invariant set of a homeomorphism. We use this informationto provide a characterization of invariant sets which admit a semi-conjugacy onto the spaceof sequences on K symbols with dynamics given by a subshift. These ideas are applied tothe Henon map to prove the existence of chaotic dynamics on an open set of parametervalues.1; Center for Dynamical Systems and Nonlinear Studies, Georgia Institue of Technology,Atlanta GA 30332.2 Research funded in part by NSF Grant DMS-91014123 Center for Dynamical Systems and Nonlinear Studies, Georgia Institue of Technology,Atlanta GA 30332; on leave from Computer Science Department, Jagiellonian University,Krak�ow, Poland. 1

1. IntroductionExamples of complicated or chaotic dynamics are ubiquitous; extending well beyondthe mathematical literature into the realm of theoretical sciences and engineering. Unfor-tunately, the set of examples for which chaos has been rigorously demonstrated is quitesmall. This is not surprising since there are at least two major obstacles hindering ver-i�cation. The �rst, and foremost is that this is a problem of global analysis. Thus, inmost cases where chaotic dynamics has been proven, the strategy has involved analyzing asimple singular or integrable problem and then perturbing the results. An added di�cultyarises from that fact that in order to insure an understanding of the dynamics after per-turbation some form of transversality is needed. (see [Mo, GH]). The second di�culty isour inability to detect and control global bifurcations. Thus, there is no simple mechanismfor determining when the chaotic dynamics of one system can be homotoped to anothersystem.These problems should be contrasted with an area of global analysis for which consid-erable progress has been made, namely, the existence theory for �xed points. The generalstrategy in this case is well known: �nd a simple system for which a �xed point can befound; determine that the degree of the �xed point is non-zero; and homotope to the sys-tem of interest. The method works because one can replace the geometric or analytic resultof the existence of a �xed point with the algebraic quantity represented by the degree. Theimportant point is that the degree is insensitive to bifurcations, i.e. as long as �xed pointsdo not encroach upon the boundary of the domain of interest the degree remains constant.This is true even if �xed points appear or disappear along the homotopy. Thus a non{zerodegree provides a lower bound (usually one) on the number of �xed points.The purpose of this paper is to present an algebraic invariant which remains constantunder appropriate homotopies and simultaneously provides a minimal description of thecomplexity of the dynamics of the system. Though, obviously, a complete understandingis the ideal result, for most applications a nontrivial lower bound on the structure of thedynamics is su�cient. With this in mind we have chosen to describe the dynamics in termsof semi-conjugacies. Estimates of this form by (implicitly given) gradient-like ows is theessence of Conley's theorem on the gradient structure of a ow (see [Co]). C. McCord2

and the �rst author proposed in [MMC] (see also [Mi], [Mi-Mo]) a technique based onthe Conley index theory, which can provide semi-conjugacies onto explicitly given systemswhich are not necessarily gradient-like. In these results all the potential chaotic dynamicswas intentionally collapsed onto periodic orbits. The point of this paper is to show thatsimilar techniques can be used to obtain semi-conjugacies with subshift dynamics on aspeci�ed set of symbols. We restrict ourselves to the case of discrete dynamical systemssince they appear to be simpler. The ow case is left for future investigation.The algebraic invariant we will use is based on the Conley index for homeomorphismsas developed in [Mr2, Mr4] (see also [RS]). Recall that since the Conley index is an index forisolated invariant sets, it remains constant through out homotopies for which an isolatingneighborhood can be preserved.Smale's horseshoe provides the perfect example with which to explain our ideas. Letf : R2 ! R2 be a plane homeomorphism which transforms the square Q with verticesA;B;C;D into the horseshoe-shaped region A0B0C 0D0 as indicated in Figure 1. The goalis to describe the dynamics on the maximal invariant set in Q, which is denoted by Inv(Q).Let the two components of f�1(Q) \ Q be given by N0 and N1, and set N := N0 [ N1.Observe that Inv(N) = Inv(Q). Thus given x 2 Inv(Q) its trajectory lies in N = N0 [N1,and hence, there is a well de�ned sequence of the symbols 0 and 1 which indicate howx travels through Q. This of course de�nes a map � : Inv(Q) ! �2, the space of bi{in�nite sequences consisting of 0 and 1. For this example it is easy to check that � is ahomeomorphism, and that if � : �2 ! �2 denotes the shift map then� � f = � � �i.e. � is a conjugacy.Our aim is to use the Conley index to obtain algebraic invariants which guarantee theexistence of a surjective map �. The �rst step in this direction is to identify the appropriateisolating neighborhoods. Consider the four componentsN00;N01;N10 andN11 of N\f(N)as indicated in Figure 1. It is easy to check that any combination of these sets forms anisolating neighborhood. Furthermore:� Inv(N00) and Inv(N11) consist of �xed points whose symbolic descriptions are given3

by the constant bi{in�nite sequences of 0 and 1, respectively.� Inv(N01) = ; = Inv(N10). Any point whose trajectory leaves N00 and enters N11passes through N01. Similarly, trajectories from N11 to N00 pass through N10.� (Inv(N00); Inv(N11)) forms an attractor{repeller pair (in the sense of Conley [Co]) forInv(N11 [N10 [N00); as does (Inv(N11); Inv(N00)) for Inv(N00 [N01 [N11).

Figure 1. U-horseshoeThis last point brings us to the issue of connecting orbits for attractor-repeller decom-positions of an isolated invariant set. For both ows [Co] and maps [Mr3], it is known thatthe non{triviality of the boundary homomorphism for the exact sequence of the attractor{repeller pair implies the existence of connecting orbits. However, being a boundary map itcan only detect the existence of connections when the di�erence in the index is of degree1. In the case of the horseshoe the index of Inv(N00) is isomorphic to that of Inv(N11)4

and hence the boundary isomorphism must be zero. For maps, however, the index consistsof both a graded module and an automorphism on this module. We show (Theorem 2.2)that the index automorphism for Inv(N00 [N01 [N11) can carry information concerningthe existence of connecting orbits from Inv(N00) to Inv(N11).As will be explained later these automorphisms can be viewed as matrices where thebasis of the modules can be expressed in terms of the indexes of Inv(N00) and Inv(N11).In the simplest setting, i.e.Con�(Inv(N00)) � Con�(Inv(N11)) � Z2;there exist homomorphisms �kl which map the index of Inv(Nll) to the index of Inv(Nkk).Thus a composition of these isomorphism, e.g.�11 � �10 � �01 � �11 � �10is well de�ned and the proof (presented in Section 6) of the main theorem (Theorem 2.3) isnow reduced to showing that if each �kl is an isomorphism then this implies the existenceof a set of orbits which pass through the sets N0 and N1 in the orderN1 ! N1 ! N0 ! N1 ! N1 ! N0:To do this we need to make observations concerning the Alexander{Spanier cohomologytheory (Section 3).However, at the level of generality considered in this paper there is no naturally de�nedhomomorphism �kl. While the explanation of this lack of naturality is very simple, itsimportance cannot be overstated. The algebra of the Conley index is obtained from thetopology of the dynamics via the direct limit functor L (see Section 4). This involves a lossof information. On the other hand, it is the index which continues rather than the topology.Therefore, from the point of view of applicability, the assumptions and hypothesis mustbe stated in terms of the algebra, while the conclusions should be stated in terms of thedynamics. Thus one of the essential problems is how does one lift the algebraic informationback to the topological or geometrical information.5

Returning to the problem at hand, concerning connecting orbits from Inv(Nll) toInv(Nkk), we conjecture that the relevant information fromCon�(Inv(Nll)) to Con�(Inv(Nkk))is carried by the linear conjugacy classes of the homomorphisms between the spaces. Inthis paper we circumvent this problem by introducing the connection number �lk whichreplaces the homomorphism �kl in the argument above.Before turning to the next section, where precise statements of our results can befound, we wish to emphasize that the geometric ideas which underlie our algebraic tech-niques are by now well understood. A careful description of how geometric and analyticmethods can be used to construct a conjugacy � can be found in Moser1 and leads to twoobservations:1. the general idea for constructing the map onto the sequence of symbols involves show-ing that the forward and backward images of carefully chosen subsets of phase spaceintersect in a transverse fashion;2. the homeomorphism is obtained by imposing conditions on the rates of contractionand expansion of the sets.Implicit in the existence of our isomorphisms between the graded modules of theConley indexes is some form of transversality. The advantage to the algebraic approach isthat this transversality whether topological or di�erential need not be checked explicitly.On the other hand, eliminating the use of analysis precludes an estimate on the expansionand contraction rates, and hence, there is no hope of strengthening the semi-conjugacy toa conjugacy, i.e. we have explicitly given up on an exact description of the dynamics.One �nal comment is in order. The Conley index of an isolated invariant set is com-puted by choosing an isolating neighborhood and quotienting the exit set to a point. Obvi-ously, information is lost in this process. In discussions Conley emphasized the importanceof understanding how to capture this information. (See [Co II.2.4] for an illuminating ex-ample and [E2] for a concrete description in the setting of maps). In the setting of discretedynamical systems our results can be viewed as a partial answer to this problem, thoughin the case of ows the problem remains essentially untouched.1 For a description of the geometric ideas explicitly stated in terms of isolating neigh-borhoods and how this leads to the mapping � see [E1].6

To explain what we mean by this, let us return to the example of the Smale horseshoe.As presented above, Q is the natural isolating neighborhood for the horseshoe. However,the Conley index of Inv(Q) is trivial from which one can draw no conclusions concerningeven the existence of an invariant set for Q. The results of this paper imply the following:subisolating neighborhoods and their indexes provide information about the dynamics onthe global isolated invariant set. Obviously, given an in�nite sequence of isolating neigh-borhoods which converge down onto the invariant set and knowledge of the various indexes,one would expect to be able to provide a highly detailed description of the dynamics. How-ever, obtaining this information directly appears to be unreasonably di�cult for nontrivialsystems (one can interpret Theorem 2.4 as a means of obtaining such information, butby continuing the ow to an easily understood example where the complete informationis known). Furthermore, and this is the point of this paper (Theorem 2.3), non-triviallower bounds on the complexity of the dynamics can be obtained from a �nite amount ofinformation. An open question is how much redundancy exists in the index informationthat can be computed, i.e. do there exist relationships between the various indexes sothat non-triviality of some indexes forces others to be non-trivial (like in the case of Morsedecompositions for which one has the existence of connection matrices).The organization of the paper is as follows. Our main results are presented in Sec-tion 2. In Section 3 we brie y recall some basic facts concerning the Alexander-Spaniercohomology. Isolating neighborhoods and the Conley index for maps are summarized inSection 4. Proofs of preliminary results and of the main theorems are presented in Sections5 and 6. (The reader not familiar with the Conley index theory will probably prefer tostudy the sections in the order 1,3,4,2,5,6,7.) Finally, we conclude in Section 7 with anexample which is meant to indicate:1. how the Conley indexes can be computed for a speci�c dynamical system, and2. the need for additional theorems of the type presented here.The dynamical system we have chosen to use is the Henon map. Though we believethat the results presented here are new, it must be emphasized that it is the techniques(both the strengths and weaknesses, thereof) rather than the results which are the focusof this section. As we hope will be made clear, it is premature to believe that a systematic7

study of the Henon attractor should be undertaken with the current set of index tools.

8

2. Main results.Let f : X ! X be a homeomorphism of a locally compact metric space X. N � Xwill always denote an isolating neighborhood under f which decomposes into K disjointcompact subsets fNk j k = 1; : : : ;Kg. LetA = h�ijibe a K �K matrix with entries �ij 2 f0; 1g. De�ne�A � 1Y�1f1; : : : ;Kgby �A = fa = (: : : ; a�1; a0; a1; : : :) j �anaan+1 = 1gand let � : �A ! �A be given by (�(a))n = an+1:Let �nA � �A denote the collection of all subsequences of length n in �A and let �nKdenote all subsequences of length n on K symbols. Given a = (a0; : : : ; an) 2 �nA setNa := n\i=0 f i(Nai )and given a = (a0; : : : ; an) 2 �nK n �nA set Na = ;. For any A � �nK de�neNA := [a2ANa:To simplify the notation we shall setSA := Inv(NA):The following result is trivial. However, because of its importance we elevate it to thestatus of a theorem. 9

Theorem 2.1. For any A � �nK the set NA is an isolating neighborhood for f .For each isolating neighborhood NA its Conley indexCon�(NA) = (CH�(NA); ��(NA))is de�ned. Recall [Mr], that it is a pair consisting of a graded module CH�(NA) over a �xedcommutative ring with unity G and an isomorphism ��(NA) : CH�(NA)! CH�(NA).It follows from Theorem 2.1 that Nk;Nl and Nfkk;kl;llg are isolating neighborhoods.This sets up the following theorem which relates the index automorphism to the existenceof connecting orbits.Theorem 2.2.(i) (Sk; Sl) is an attractor-repeller pair in Sfkk;kl;llg.(ii) The Conley indices of Sk; Sl and Sfkk;kl;llg are related by the following commutativediagram with exact rows(1) 0 �! CH�(Sl) �! CH�(Sfkk;kl;llg) �! CH�(Sk) �! 0??y�(Sl) ??y�(Sfkk;kl;llg) ??y�(Sk)0 �! CH�(Sl) �! CH�(Sfkk;kl;llg) �! CH�(Sk) �! 0in which vertical arrows denote the corresponding index maps.(iii) (2) CH�(Sfkk;kl;llg) = CH�(Sk)� CH�(Sl):(iv) If(3) �(Sfkk;kl;llg) and �(Sk) � �(Sl) are not conjugatethen there exists a connecting orbit from Sl to Sk in Sfkk;kl;llg, i.e. an elementx 2 Sfkk;kl;llg such that �(x) � Sk and !(x) � Sl.Condition (3) is equivalent to assuming that every maximal non-decomposable in-variant subspace of �(Sk) when embedded into CH�(Sfkk;kl;llg) is also maximal with re-spect to �(Sfkk;kl;llg). This can be veri�ed by comparing the Jordan forms of �(Sk) and�(Sfkk;kl;llg). 10

Since both isomorphisms in (3) are graded isomorphisms, they fail to conjugate if theydo so in one of the components. Thus consider the special case when for some �xed m 2 N(4) Conm(Sl) � Conm(S1) � (G; id):It follows then from (1) that in matrix form�k(Sfkk;kl;llg) = � 1 �0 1 �in which � can depend on the choice of the complement to CHm(Sk) in the decomposition(2) given by (1). However, since all such matrices which represent the same map areconjugate, the number �kl := � 0 if � = 01 if � 6= 0is an invariant of the attractor-repeller pair (Sk; Sl) in Sfkk;kl;llg.We shall call �kl a connection number, because �kl 6= 0 implies �m(Sfkk;kl;llg) and�m(Sk) � �m(Sl) are not conjugate. Hence, by Theorem 2.2, if �kl = 1, then there existsa connecting orbit from Sl to Sk (compare with Example 5.5 in [RSZ]).The G-horseshoe as presented in Figure 2 satis�es condition (4) with m = 1 and itis easy to verify that both connection numbers �12 and �21 are one. The U-horseshoe inFigure 1 fails to satisfy (4) when the coe�cients are rationals, because then �1(S1) = � id.However it does satisfy (4) when we take Z2 coe�cients. Hence also in this case we havewell de�ned connection numbers �21 = �12 = 1.Theorem 2.2 illustrates that some information is lost when the discrete Conley indexis considered as a zeta function (see [RSZ]) or as a collection of the eigenvalues of �(S).The following theorem provides the semi-conjugacy and the subshift dynamics usedto describe Inv(N).Theorem 2.3. Let N be an isolating neighborhood. Assume that N is the disjoint unionof compact sets fNk j k = 1; : : : ;Kg and that for k = 1; : : : ;KConm(Sk) � Conm(Inv(Nk)) � (G; id):11

Let �kl denote the connection number obtained from Nfkk;kl;llg and set �kk = 1. De�neA = h�kli to be the resultingK�K matrix. If �kl = 0 implies that there are no connectingorbits from Sl to Sk, then there exists d 2 N and a continuous surjection� : Inv (N)! �Asuch that the following diagram commutesInv(N) fd�! Inv(N)??y� ??y��A ��! �A

Figure 2. G-horseshoeThe horseshoes in Figure 1 and in Figure 2 were models for Theorem 2.3. To applythis theorem to those examples, let K = 2 and observe thatA = � 1 11 1 � :12

This gives the desired semi-conjugacy onto the full shift dynamics on two symbols.Remark: The assumption �kl = 0 implies that there are no connecting orbits from Slto Sk in Theorem 2.3 is necessary, and shows that the algebraic information is not alwayssu�cient. This may pose a serious obstacle in some applications. On the other hand,as was shown for the horseshoe, in the case of a full shift this assumption is vacuouslytrue. Furthermore, the hypothesis that connecting orbits do not exist within an isolatingneighborhood is fundamentally di�erent from assuming the existence of connecting orbits.To show that connecting orbits do not exist it is su�cient to show that a section to theunstable set of the repeller is mapped out of the isolating neighborhood after a �nitenumber of iterations. Thus, it is conceivable that for speci�c problems this condition couldbe veri�ed.The fact that d must be chosen from the set of natural numbers, rather than the\optimal" d = 1, arises from algebraic considerations in the proof. What this means geo-metrically is still not clear. In particular, the minimal possible d appears to depend uponan interplay between the subshift with which one wishes to de�ne the semi{conjugacy andthe labeling of the components of the isolating neighborhood. Under stronger assumptionsone can show that d = 1 and that the pre-image of periodic orbits in the shift dynam-ics contains true periodic orbits of the original dynamical system. As an example of thisconsider the following theorem.Theorem 2.4. Assume M � Rn = R2 � Rn�2 is a disjoint sum of two its compactsubsets M0;M1, Ni := Mi \R2 and ft : Rn ! Rn, (t 2 [0; 1]) is a homotopy. If N is anisolating neighborhood for all ft and f0 is a U- (or G-) horseshoe in Rn, i.e. it decomposesinto a homeomorphism with a globally attracting �xed point in Rn�2 and a U- (or G-)horseshoe as in Fig.1 or in Fig.2 in R2 then there exists a continuous surjectionp : InvN ! �2such that the following diagram InvN f1�! InvN??yp ??yp�2 ��! �213

commutes. Moreover, for every periodic sequence � 2 �2 the set p�1(�) contains a periodicpoint of f1 with the same period as �.Assumptions of Theorem 2.4 are essentially stronger than those in Theorem 2.3. Tosee this consider a homeomorphism R2 � S1 ! R2 � S1 de�ned as a product of the G-horseshoe homeomorphism in Fig. 2 and the identity on S1. Take Mi := Ni � S1, whereNi; (i = 0; 1) is de�ned as in Figure 2. Then a direct computation shows that(5) Conn(M0) = Conn(M1) = � 0 for n =2 f1; 2g(G; id) for n 2 f1; 2gand both connection numbers can be easily shown to be one. Hence the conclusion ofTheorem 2.3 applies to f and also to any small perturbation of f . However (5) shows thatthere exists no homotopy to a G-horseshoe (or U-horseshoe) so that Theorem 2.4 is notapplicable. The above example also shows that the assumptions of Theorem 2.3 cannotimply the existence of periodic points (replace the identity on S1 with irrational rotation).We �nish this section with the following conjecture.Conjecture 2.5. Let N = N1 [ N2 be an isolating neighborhood where N1 and N2 aredisjoint compact sets. Assume thatConm(Sk) � Conm(Inv(Nk)) � (G; id):and that Con�(Inv(N)) � 0:Then, there exists d 2 N and a continuous surjection� : Inv (N)! �Asuch that the following diagram commutesInv(N) fd�! Inv(N)??y� ??y��A ��! �A14

3. Alexander{Spanier CohomologyIn this section de�nitions and auxiliary results concerning the Alexander{Spanier co-homology are summarized. LetG be a �xed commutative ring with unity. For a topologicalspace X and n 2 N, let �n(X) denote the G-module of all functions ' : Xn+1 !G. Thecoboundary homomorphism � : �n(X)! �n+1(X) is de�ned by the formula�'(x0; : : : ; xn+1) = n+1Xi=0(�1)i'(x0; : : : ; xi�1; xi+1; : : : ; xn+1):Since �� = 0, ��(X) := (�n(X); �) is a cochain complex overG. For ' 2 �n(X) de�ne thesupport of ' byj'j := fx 2 X j 8 V � X open x 2 V ) 9 x0; : : : ; xn 2 V s.t. '(x0; : : : ; xn) 6= 0g:Then, �n0 (X) := f' 2 �n(X) j j'j = ;g is a submodule of �n(X) and ��0(X) := (�n0 (X); �)is a cochain subcomplex of ��(X). The quotient cochain complex ��(X)=��0(X) is denotedby ���(X). If u 2 ���(X) then j'j does not depend on ' for all representations ' 2 u. Henceone can set juj := j'j, where ' 2 u.If Y is another topological space and g : X ! Y is continuous then there is an inducedcochain map g# : ��(Y )! ��(X) de�ned by the formula(g#')(x0; : : : ; xn) = '(g(x0); : : : ; g(xn)):Since g#(��0(Y )) � ��0(X), there is also an induced map g# : ���(Y )! ���(X).If i : X0 ,! X is an inclusion, then ���(X;X0) := ker i# is a cochain subcomplex of���(X). The Alexander{Spanier cohomology of the pair (X;X0) is de�ned as the homologyof the cochain subcomplex ���(X;X0) and denoted by �H�(X;X0). If (Y; Y0) is anothertopological pair and � : ���(Y; Y0)! ���(X;X0) is a cochain map then it induces the map�� : �H�(Y; Y0) ! �H�(X;X0). In particular if g : (X;X0) ! (Y; Y0) is a continuous mapthen one has a map (g#)� : �H�(Y; Y0)! �H�(X;X0), which for the sake of simplicity willbe denoted by g�.The following four propositions are easy to verify.15

Proposition 3.1. (see [Ma,p.229]) Assume ' 2 ��(X). Thenf(jf#(')j) � j'j:Proposition 3.2. (see [Ma,p.227]) Assume ' 2 ��(X). Then['] 6= 0 ) j'j 6= ;:Assume X0;X1 � X and ' 2 �p(X;X0 \X1). For i = 0; 1 de�ne�Xi(')(x0; : : : ; xp) := n'(x0; : : : ; xp) if x0; : : : xp 2 X1�i0 otherwiseProposition 3.3. If X0;X1 are closed subsets of X then �Xi : ��(X;X0 \ X1) 3 ' !�Xi(') 2 ��(X;Xi) is a well de�ned cochain map which also induces a cochain map��Xi : ���(X;X0 \X1)! ���(X;Xi)Proposition 3.4. Assume V0; V1; V areG-modules �Vi : Vi ! V , pVi : V ! Vi for i = 0; 1are linear maps such that(6) pVi � �Vj = � idVj i = j0 i 6= jand(60): �V0 � pV0 + �V1 � pV1 = idVThen(7) (�V0 � �V1) : V0 � V1 ! Vis an isomorphism with inverse (pV0 ; pV1).Proposition 3.5. Assume X0;X1 � X are closed subsets of X such thatX = intX0 [ intX1:16

If �Xi : (X;X0 \X1)! (X;Xi) (i = 0; 1) denote inclusions, then�#Xi : ���(X;Xi)! ���(X;X0 \X1)are also inclusions and ���(X;X0 \X1) = ���(X;X0) � ���(X;X1)with the corresponding projections ��X0 and ��X1. Moreover�H�(X;X0 \X1) �= �H�(X;X0)� �H�(X;X1)with the corresponding inclusions ��X0; ��X1 and projections ��X0 ; ��X1. Furthermore, if Y0; Y1are closed subsets of Y such that Y = intY0 [ intY1 and� : ���(X;X0 \X1)! ���(Y; Y0 \ Y1)is a cochain endomorphism then � decomposes into a matrix of cochain maps� = ��00 �01�10 �11 � ; where �ij = ��Yi � � � �#Xj :Also, �� = ���00 ��01��10 ��11 � ; where ��ij = ��Yi � �� � ��Xj :Proof. The assumptions of Proposition 3.4 need to be veri�ed. First, consider (6). Fixi; j 2 f0; 1g, take ' 2 �p(X;Xj ) and set'ij := �' if i = j0 if i 6= j.It needs to be shown that �Xi' � 'ij is locally zero. Choose x 2 X. If x 2 intXj thenthere exists a neighborhood V of x such that V � Xj and 'jV = 0. Since (�Xi' �'ij)(x0; : : : ; xp) 2 f�'(x0; : : : ; xp); 0g, it follows that (�Xi' � 'ij)jV = 0. Thus, assumex 2 intX1�j . Choose a neighborhood V of x such that V � X1�j . One can modify V sothat, in addition, 'jV\Xj = 0 if x 2 Xj and V \Xj = ; if x =2 Xj . Then, �Xj'jV = 'jV .Furthermore, if i = j then 'iijV = 'jV , and hence, �Xj'jV � 'jV = 0. Assume i 6= j and17

choose x0; : : : ; xp 2 V . If x0; : : : ; xp 2 Xj then (�Xi'�'ij)(x0; : : : ; xp) = '(x0; : : : ; xp) =0; otherwise (�Xi')(x0; : : : ; xp) = 0. Therefore, (�Xi'� 'ij)jV = 0.Take in turn � 2 �p(X;X0 \X1). Then,h��#X0 � �#X1� � (�X0 ; �X1)i (�) = �X0(�) + �X1(�):In order to prove (60) it needs to be shown that �X0(�) + �X1(�) � � is locally zero. Tothis end take x 2 X. Assume �rst that x 2 intX0. Choose V a neighborhood of x suchthat V � X0. Without loss of generality it can also be assumed thatV \X1 = ; if x =2 X1or �jV\X0\X1 = 0 otherwise:Choose x0; : : : ; xp 2 V . Then�X0(�)(x0; : : : ; xp) = 0 and �X1(�)(x0; : : : ; xp) = �(x0; : : : ; xp);hence (�X0(�) + �X1(�) � �)jV = 0:A similar argument in case x 2 intX1 �nishes the proof of (60). The remaining part of theproposition is obvious.18

4. Isolated Invariant Sets and the Conley Index for MapsAssume X is a locally compact metric space and f : X ! X is a homeomorphism. Asubset S � X is called invariant (with respect to f) if f(S) = S. The maximal invariantsubset of a given set N � X will be denoted by Inv(N; f) or Inv N , i.e.InvN = fx 2 N j fn(x) 2 N for all n 2 Zg:Similarly one de�nes InvN+ = fx 2 N j fn(x) 2 N for all n 2 Z+g and InvN� = fx 2N j fn(x) 2 N for all n 2 Z�g. A compact set N � X is called an isolating neighborhoodif Inv N � intN . The following statement is a triviality.Proposition 4.1. If N is an isolating neighborhood then also for any n 2 N,n\i=0 f i(N)is an isolating neighborhood and Inv (Tni=0 f i(N)) = Inv N .Assume N is an isolating neighborhood.De�nition 4.2. P = (P1; P2) will be called an index pair for f in N if P1; P2 are closedsubsets of N with P2 � P1 and(i) Pi \ f�1(N) � f�1(Pi), i = 1; 2(ii) P1nP2 � f�1(N)(iii) Inv N � int(P1nP2)Additionally, if(iv) cl(P1nP2) \ cl(f(P2)nP1) = ;, we call the index pair P regular.The following proposition follows easily from the proof of Lemma 4 in [Sp, p. 309]Proposition 4.3. If P is a regular index pair then iP : (P1; P2)! (P1[f(P2); P2[f(P2))induces an isomorphism i# : ���(P1 [ f(P2); P2 [ f(P2))! ���(P1; P2) and(i#;pP )�1([']) = [�P (')]where �P : ��(P1; P2)! ��(P1 [ f(P2); P2 [ f(P2)) is de�ned by�P (')(x0; : : : ; xp) = n'(x0; : : : ; xp) if x0; : : : ; xp 2 P10 otherwise.19

De�nition 4.4. The cochain map f# � (i#)�1 will be called the cochain index map andwill be denoted by f#P . The cohomology endomorphism f�P := (f#P )� will be called the(cohomology) index map.The following de�nition of an isolating block for maps was proposed in [E2].De�nition 4.5. The compact set N � X is called an isolating block for f if f�1(N) \N \ f(N) � intN .Proposition 4.6. Assume N is an isolating neighborhood and M � N is both open andclosed in N . Then also M is an isolating neighborhood. Moreover, if N is an isolatingblock then so is M .Proof. Under the assumptions of our proposition we have Inv M � M \ Inv N �M \ intN =M \ (intM [ int(NnM)) = intM . If N is an isolating block then f�1(M) \M \ f(M) �M \ intN = intM .For an isolating block N we de�ne N� := Nnf�1(intN) and call it the exit set.Proposition 4.7. Each isolating block is an isolating neighborhood. If N is an isolatingblock then (N;N�) is a regular index pair for f in N .Proof. Inv N = T1k=�1 fk(N) � T1k=�1 fk(N) � intN , hence N is an isolating neigh-borhood. Obviously N�;N are compact subsets of N and N� � N . To show thatN� \ f�1(N) � f�1(N�) take x 2 N� \ f�1(N) and assume f(x) =2 N�. Then f2(x) 2intN , i.e., f(x) 2 f�1(N) \N \ f(N) � intN and x =2 N�, a contradiction. This provesproperty (i). Property (ii) is obvious. To see (iii) observe that f(Inv N) = Inv N � intN ,thus Inv N � intN \ f�1(intN) = int(N \ f�1(intN)) = int(NnN�):Finally observe that cl(f(N�)nN) \ cl(NnN�) � cl(f(N)n intN) \ cl(N \ f�1(intN))� (f(N)n intN)\(N \f�1(N)) � (f(N)n intN)\(f(N)\N \f�1(N)) � (f(N)n intN)\intN = ;. This shows that (N;N�) is a regular index pair.The following proposition is also proved in [E2]. We include the proof for sake ofcompleteness. 20

Proposition 4.8. If S is an isolated invariant set then for every neighborhood V of Sthere exists N � V an isolating block for S.Proof. By [Mr1, Th. 2] there exist U an open neighborhood of S and continuous functions'; : U ! [0;1) such that'(x) > 0; x 2 U \ f�1(U) ) '(f(x)) < '(x); (x) > 0; x 2 U \ f�1(U) ) (f(x)) > (x);S � '�1(0) \ �1(0)and for every neighborhood W of S there exists � > 0 such thatM� := clfx 2 U j ('(x) < �; (x) < �g �W:Select � > 0 such thatM� � U\f�1(U)\V . We will show thatM�\f(M�)\f�1(M�) �intM�. For this end take x 2 M� \ f(M�) \ f�1(M�). Then '(x) = '(f(f�1(x))) <'(f�1(x)) � � and (x) < (f(x)) � �. Thus the inequalities hold also in some neighbor-hood of x and consequently x 2 intM�.Since (N;N�) is an index pair in N , a corresponding cochain index map and coho-mology index map are de�ned. We will denote them by f#N and f�N respectively. Theinverse of the inclusion i# : ���(N [ f(N�);N� [ f(N�)) ! ���(N;N�) will be denotedby �N , so that f#N = f# � �Nf�N = f� � ��N :In the sequel we will also need the following simple proposition.Proposition 4.9. f(Nn intN N�) � N .A subset A � S is called an attractor (repeller) in S i� there exists a neighborhood Uof A in S such that the !-limit set (�-limit set) of any point x 2 U is contained in A. Onecan verify (see [Mr3]) that if A+ � S is an attractor in S then the set A� of points in Swhich have their positive limit set disjoint from A+ is a repeller in S. The pair (A�; A+)is called a repeller-attractor pair in S. 21

Assume A�; A+ are two disjoint invariant subsets of S. The set of connections fromA� to A+ in S, Conn(A�; A+;S) is de�ned as the set of those x 2 S whose positive limitset is in A+ and negative limit set is in A�.One can easily verify the following proposition.Proposition 4.10. If A�; A+ are two disjoint invariant subsets of S then (A�; A+) is arepeller-attractor in S i� A� [A+ [ Conn(A�; A+;S) = S:By Propositions 4.8 and 4.9 every isolated invariant set S admits an index pair (P1; P2)in some isolating neighborhood N which isolates S. The pair (H�(P1; P2); f�P ) is an ob-ject in Endo(G-GMod), the category of endomorphisms of graded modules over G (see[Mr4]). There is a natural functor from this category to its subcategory of automorphismsAuto(G-GMod). It assigns to every endomorphism e : E ! E the automorphism inducedon the direct limit of the sequence E e�!E e�!E e�! : : :We call it the direct limit functor and denote it by L. It turns out that L(H�(P1; P2); f�P )does not depend on the index pair (P1; P2) (see [Mr4, Th. 1.7]), hence it is an invariantof N and S. We call it the Conley index of the isolating neighborhood N (or the iso-lated invariant set S) and denote it Con�(N; f) (or respectively Con�(N;S)). Note thatCon�(N; f) is a graded module with a distinguished graded automorphism. We will denotethe nth component of Con�(N; f) by Conn(N; f).Functors other than L can be used in the construction of the Conley index (see [Mr4]).We use L, because of the following useful proposition.Proposition 4.11. (see [Ma, Th. A.7]). L preserves exact sequences.22

5. Auxiliary Lemmas.Lemma 5.1. Assume N is an isolating block, N0;N1 � N are compact, N = N0 [ N1and N0 \ N1 = ;. Then N0 and N1 are isolating blocks. Moreover if E0 := N� [ N1,E1 := N� [N0 then(9) ���(N;N�) = ���(N;E0)� ���(N;E1)�H�(N;N�) = �H�(N;E0)� �H�(N;E1):Furthermore, if(90) f�N = � f�N (0; 0) f�N (0; 1)f�N (1; 0) f�N (1; 1) �denotes the corresponding decomposition of the index map f�N into partial cochain maps,then for i = 0; 1(10) Con(Ni; f) = L(H�(N;Ei); f�N (i; i)):Finally, if f�N (i) denotes the map f�N with the (1� i)th row in (9') replaced with zero thenalso(11) Con(Ni; f) = L(H�(N;N�); f�N (i)):Proof. N0 and N1 are isolating blocks by Proposition 4.6. Formula (9) follows immedi-ately from Proposition 3.5. Since proofs of (10) and (11) for i = 0; 1 are analogous, withoutloss of generality we can assume that i = 0. Put g := f#N (0; 0) andE00 := E0 [ (N0 \ f�1(N1)); E01 := E0 [ (N0 \ f�1(N0)):Observe that N0 = (N0nf�1(N0)) [ (N0nf�1(N1))and N0nf�1(Ni) � N� [ (N0 \ f�1(N1�i)); i = 0; 1:23

HenceN =N1 [N0 = N1 [ (N0nf�1(N0)) [N1 [ (N0nf�1(N1))� intN (N� [N1 [ (N0 \ f�1(N1))) [ intN (N� [N1 [ (N0 \ f�1(N0)))= intN E00 [ intN E01 � N;and we obtain from Proposition 3.5 that���(N;E0) = ���(N;E00)� ���(N;E01):We claim that the corresponding decomposition of g is(12) g = � g(0; 0) g(0; 1)g(1; 0) g(1; 1) � = � g(0; 0) g(0; 1)0 0 � ;i.e. the bottom row in the decomposition is zero.To see this take ['] 2 ���(N;E0). By Proposition 3.5 we have to show that'0 := ��E01 � �E0 � f# � �N � i#E0� (')is locally zero for all ' 2 ��(N;E0).>From the de�nitions of the maps �E01 ; �E0 ; �N it follows that(13) '0(x0; : : : ; xp) = �'(f(x0); : : : ; f(xp)) if x0; : : : ; xp 2 E00 \E1 \ f�1(N)0 otherwise.Take x 2 N . Since '0 2 ��(N;E01), '0 is zero in some neighborhood of every point inintN E01. If x 2 f�1(intN E0) then we can take a neighborhood U of f(x) such thatU � E0 and 'jU = 0. Hence if V is a neighborhood of x such that f(V ) � U then'0jV = 0.Thus, it remains to consider the case x 2 Nn(intN E01 [ f�1(intN E0)). In particularwe have then x 2 N0, x =2 intN N�, f(x) =2 intN N�, f(x) =2 N1. By Proposition 4.9f(x) 2 N , thus f(x) 2 N0. We will show that x =2 N�. If x 2 N� then x 2 Nn intN N�and by Proposition 4.9 f(x) 2 N . Applying it again we �nd that also f2(x) 2 N . Thusf(x) 2 f�1(N) \ N \ f(N) � intN . It follows that x =2 N�, a contradiction. Hencex =2 N� and since x 2 N0 we see that x =2 E0. Take V a neighborhood of x such that24

V \E0 = ; and f(V ) � N0. Let x0; : : : ; xp 2 V . For every i = 0; 1; : : : ; p there is xi =2 E00because otherwise we get from xi =2 E0 that f(xi) 2 N1. It follows now from (13) that'0jV = 0 which proves that '0 is locally zero.Now observe thatN�0 = N0nf�1(intNnN1) = (N0nf�1(intN)) [ (N0 \ f�1(N1)) = N0 \E00:Hence the inclusion i0 : (N0;N�0 ) ,! (N;E00) is an excision of N1 and since clNN1 =N1 = intN N1 � intN E00, it induces an isomorphism i#0 : ���(N;E00) ! ���(N0;N�0 ) (see[Sp, p. 309]). We will show that the following diagram(14) ���(N;E00) g(0;0)����������������! ���(N;E00)??yi#0 ??yi#0���(N0;N�0 ) �N0�! ���(N0 [ f(N�0 );N�0 [ f(N�0 )) f#�! ���(N0;N�0 )commutes.To this end take ['] 2 �p(N;E00) and set 1 := (i#0 � g(0; 0))('); 2 := (f# � �N0 � i#0 )('):Observe that 1 = (i#0 � �E00 � �E0 � f# � �N � i#E0 � i#E00)('). Thus a straightforwardcomputation shows that 1(x0; : : : ; xp) = �'(f(x0); : : : ; f(xp)) if x0; : : : ; xp 2 N0 \ E01 \ f�1(N)0 otherwise.Similarly we �nd that 2(x0; : : : ; xp) = �'(f(x0); : : : ; f(xp)) if x0; : : : ; xp 2 N0 \ f�1(N0)0 otherwise.We need to show that := 2 � 1 is locally zero. Thus take x 2 N0. Obviously disappears around points x 2 intN0 N�0 , because 2 �p(N0;N�0 ). Thus we can assumethat x =2 intN0 N�0 and similarly that f(x) =2 intN E00. It follows from Proposition 4.9that f(x) 2 N0. In particular f(x) =2 N1. Choose V a neighborhood of x0 in N0 suchthat f(V ) \ N1 = ; and take x0; : : : ; xp 2 V . If for some i = 0; : : : p f(xi) =2 N then 1(x0; : : : ; xp) = 2(x0; : : : ; xp) = 0, so (x0; : : : ; xp) = 0. Otherwise f(xi) 2 N0 for25

all i = 0; : : : ; p. In particular x0; : : : ; xp 2 N0 \ f�1(N0) � N0 \ E01 \ f�1(N) and 1(x0; : : : ; xp) = 2(x0; : : : ; xp) = '(f(x0); : : : ; f(xp)), i.e., (x0; : : : ; xp) = 0. This showsthat jV = 0 and completes the proof of the commutativity of diagram (14).It follows from Proposition 3.5 and the structure of g� given by (12) that the followingdiagram with exact rows(15) 0 �! H�(N;E00) ��E00�! H�(N;E0) ��E01�! H�(N;E01) �! 0??yg�(0;0) ??yg�=f�N (0;0) ??y00 �! H�(N;E00) �! H�(N;E0) �! H�(N;E01) �! 0commutes. Proposition 4.11 implies that the sequence0 �! L(H�(N;E00); g�(0; 0)) �! L(H�(N;E0); g�) �! L(H�(N;E01); 0) �! 0is exact. However obviously L(H�(N;E01); 0) = 0, thus L(H�(N;E00); g�(0; 0)) andL(H�(N;E0); g�) are isomorphic. Now, applying the direct limit functor to diagram (14)we conclude that L(H�(N;E00); g�(0; 0)) and L(H�(N0;N�0 ); f�N0) are isomorphic. SinceL(H�(N0;N�0 ); f�N0) = Con(N0) by de�nition, the proof of (10) is �nished.Finally an argument analogous to that used with diagram (15) shows that also (11)is satis�ed.Proposition 5.2. Assume F : V ! V is a linear endomorphism of a G-module V andL(V;F ) = (G; id). Then 1 is an eigenvalue of F and for every v 2 V there exists k 2 Nsuch that either F kv = 0 or F kv is a 1{eigenvector of F . In particular the generalized1{eigenspace of F is one-dimensional.Proof. The de�nition of the direct limit implies that we have a commutative diagramV F�! V F�! V � � �& p0 ??yp1 . p2Gand since F induces the identity on the limit we have also a family of commutative diagrams(16) V F�! V??ypi ??ypiG id�! G:26

It follows from [Ma, Prop. A.4] that(17) gkerF := [n�0 kerFn = ker pi for all i 2 Nand pi is surjective for some i 2 Z. In particular we get pi = p0 for all i 2 N. Selectw 2 V such that p0(w) 6= 0. It follows from (16) that p0(w�Fw) = 0. Hence (17) impliesFn(w � Fw) = 0 for some n 2 N, i.e. Fz = z for z := Fnw. Since w =2 ker p0, we seefrom (17) that z 6= 0. This shows that 1 is an eigenvalue of F and z is a correspondingeigenvector. We get from (16) that for any v 2 V p0(v�Fv) = 0. It follows from (17) thatfor some n 2 N, Fn(v � Fv) = 0. Hence either Fnv = 0 or Fnv is a 1{eigenvector of F .

27

6. Proofs of main resultsTheorem 2.1 follows immediately from Propositions 4.1 and 4.6.Proof of Theorem 2.2. By Proposition 4.8 one can assume that Nk;l = Nk [Nl is anisolating block.(i) The �rst step is to show that(18) Nk \ Sfkk;kl;llg � Inv+(Nk):To this end take x 2 Nk \ Sfkk;kl;llg. By de�nitionSfkk;kl;llg = Inv�(Nk \ f(Nk)) [ (Nk \ f(Nl)) [ (Nl \ f(Nl))�hence, x 2 Nk \ Inv�(Nk \ f(Nk)) [ (Nk \ f(Nl))�� Nk \ Inv(Nk):Thus f(x) 2 Nk. A similar argument proves that(19) f(Nl) \ Sfkk;kl;llg � Inv�(Nl):Observe that Sj = Inv(Nj) = Inv(Nj \ f(Nj )) � Sfkk;kl;llg for j = k; l. HenceSl [ Sk [ Conn(Sl; Sk;Sfkk;kl;llg) � Sfkk;kl;llg:The opposite inclusion follows immediately from (18) and (19). Hence Proposition 4.10shows that (Sk; Sl) is a attractor-repeller pair in Sfkk;kl;llg.(ii) To simplify the notation letM = Nfkk;kl;lk;llgand let E =M� :=M n f�1(M):28

The cohomology indices of interest for our computation are:V = �H�(M;E);Vk = �H�(M;E [Nll [Nlk);Vl = �H�(M;E [Nkk [Nkl);Vkk = �H�(M;E [Nll [Nlk [Nkl);Vll = �H�(M;E [Nkk [Nlk [Nkl);Vkl = �H�(M;E [Nll [Nlk [Nkk);Vlk = �H�(M;E [Nll [Nkk [Nkl):Lemma 5.1 applied to M , Nkl [Nkk, and Nll [Nlk implies thatV = Vk � Vl:Similarly, for i = k; l, Vi = Vik � Vil:f�M : V ! V can be decomposed as(20) f�M = �Fkk FklFlk Fll � where Fij : Vj ! Vi i; j 2 fk; lgand each Fij can be further decomposed as(21) Fij = �Bikjk BikjlBiljk Biljl �where Bimjn : Vjn ! Vim; m; n 2 fk; lg:Since f(Nij ) � Nki [Nli, one can use the argument of Lemma 5.1 which gave rise to (12)to conclude that f�M = 264Bkkkk Bkkkl 0 00 0 Bkllk BklllBlkkk Blkkl 0 00 0 Blllk Bllll 375 :Applying (20) and Lemma 5.1 to M;Nkl [Nkk;Nlk [Nll results in(22) Con(Nk) = L�V; �Fkk Fkl0 0 �� = L(Vk; Fkk)29

and(23) Con(Nl) = L�V; � 0 0Flk Fll �� = L(Vl; Fll)Applying (21) and Lemma 5.1 to M;Nlk;Nkk [Nkl [Nll we �nd that(24) Con(Nkk [Nkl [Nll) = L(V; �Fkl) with �Fkl = �Fkk Fkl0 Fll �Let � : Vk ! V and � : V ! Vl denote the inclusion and projection corresponding tothe direct sum V = Vk � Vl. Then, we have the commutative diagram with exact rows(25): 0 �! Vk ��! V ��! Vl �! 0??yFkk ??y �Fkl ??yFll0 �! Vk ��! V ��! Vl �! 0Applying the functor L and using the assumptions (22), (23) and (24) gives the com-mutative diagram (1). Since, by Proposition 4.11 L preserves exact sequences, the rows in(1) are exact.(iii) follows immediately from (1) and (iv) is a consequence of the additivity propertyof the Conley index (see [Mr4]).Proof of Theorem 2.3. Let m be as in (4). We will follow the notation introduced inthe proof of Theorem 2.2 under the assumption that all graded endomorphisms Fij ; �Fijintroduced there are now restricted to the mth level. Observe that by (4) and Proposition5.2 Fii has a unique (up to a constant) 1{eigenvector vi. We will show that Fijvj =2 gkerFii(comp (17)). If Fijvj 2 gker Fii then F piiFijvj = 0 for some natural number q. Putw0 = vj ; q�1Xn=0FniiFijvj! ; w1 = (0; vi)It is straightforward to verify that w0 and w1 are linearly independent 1{eigenvectors of�Fij . Also, an easy computation shows that Dij = L( �Fij) has two linearly independent1{eigenvectors. This implies that �ij = 0, a contradiction. Hence Fijvj is not in thegeneralized kernel of Fii. By Proposition 5.2 there exists a natural number dij such thatF dijii Fijvj is a 1{eigenvector of Fii. 30

Take d := 1 +max1�i;j�Kfdi;jg. Then(26) F d�1ii Fijvj is a 1{eigenvector of Fii for any i; j 2 f1; : : : ;Kg; i 6= j:De�ne the map � : N ! f1; : : : ;Kg by�(x) := k if x 2 Nkand the map p : Inv N ! �K byp(x) := (�(f�nd(x)))n2ZObviously � and p are continuous (observe that this is the point at which the assumptionthat �kl = 0 implies Conn(Sl; Sk) = ; is used) and p � fd = � � p. It remains to show thatp is surjective. For this end take an element � : Z! f1; : : : ;Kg of �A and �x n 2 N. Let'i 2 vi be a cochain representative of vi. De�ne recursively a sequence(27) �n := '��n i := [f#;kM (�i; �i)d�1 � f#;kM (�i; �i�1)]( i�1) i = �n+ 1;�n+ 2; : : : ; n:Then i 2 ��k(N;E�i). In particular j ij � cl(NnE�i ) � N�i . From (27) and Proposition3.1 one has that f j (j ij) � N; j = 0; 1; : : : dand fd(j ij) � j i�1j i = �n+ 1;�n+ 2; : : : ; n;which implies that(28) f j(j nj) � N; j = 0; 1; : : : 2ndand(29) fmd(j nj) � N�n�m m = 0; 1; : : : ; 2n:By (26) and (27) the cohomology class of n is a 1{eigenvector of F�n�n = fkM (�n; �n).Hence this class is di�erent from zero and by Proposition 3.2 j nj 6= ;. Select yn 2 j njand put xn := fnd(yn). Then, by (28)(30) f i(xn) 2 N; i = �nd;�nd+ 1; : : : nd31

and by (29)(31) f�id(xn) = f (n�i)d(yn) 2 f (n�i)d(j nj) � N�i i = �n;�n+ 1; : : : ; n:In particular xn 2 N�0 . Since N�0 is compact, one can assume that xn ! x� 2 N�0 . From(30) one obtains x� 2 InvN and from (31)f�id(x�) 2 N�i for i 2 Z;i.e. �(f�id(x�)) = �i for i 2 Z. This shows that p(x�) = � and proves that p is surjective.Proof of Theorem 2.4. De�ne the map p as in the proof of Theorem 2.3. Let � : Z!f0; 1g be a periodic sequence in �2 with period q. By Theorem 2.1 M� is an isolatingneighborhood for all ft, (t 2 [0; 1]). In particular (ft)q has no �xed points in the boundaryof M�. Thus, the �xed point index i((ft)q; intM�) is de�ned (see [Do,7.5]) andi((f0)q ; intM�) = i((f1)q; intM�):The multiplicativity property of the �xed point index implies thati((f0)q; intM�) = i((f0jR2 ; intN�):The latter is the �xed point index of a hyperbolic �xed point, hence it is di�erent fromzero. Thus there exists x 2 InvM , a q-periodic point of f such that p(x) = �. Thesurjectivity of p follows from the fact that periodic sequences are dense in �2 and InvNis compact.32

7. The Henon MapRecall that the Henon map is given byH(x; y) = (1 + y � ax2; bx):In order to provide several examples of how our ideas can be applied to speci�c systemswe shall study the dynamics of H for the set of parameter valuesf(a; b) j a = 1:8; 0 � b � 0:025g:Simple numerical experiments suggest that for these parameter values the Henon map hasan attractor, upon which the dynamics may be rather complicated. The �rst observationwhich must be made is that Theorem 2.3 cannot provide direct information about theentire structure of an attractor. In order for �kl = 1, it must be that Inv(Nl) have anunstable \manifold". This forces the Inv(N) to be unstable. What is needed is a theorem,based on the index information, which can provide information concerning the structureof a chaotic attractor. How to do this in an applicable manner remains an open question.We avoid this problem by studying the invariant set generated by H2, the seconditerate of H, of a neighborhood eN which we shall de�ned below. Observe that proving theexistence of a semi-conjugacy from Inv( eN;H2) to a horseshoe, provides a lower bound onthe complexity of the dynamics of H.Before stating our result we introduce some notation which will be used throughoutthis section. Let p1 = �0:54198 p2 = �0:403384p3 = �0:348221 p4 = �0:178249p5 = 0:178249 p6 = 0:348221p7 = 0:403384 p8 = 0:54198and set N = 4[i=1 [p2i�1; p2i]:33

We shall make use of the following functions:h1(x; y) := 1 + bx � a(1 + y � ax2)2h2(x; y) := b(1 + y � ax2)h�11 (x; y) := x� 1b + ay2b3h�12 (x; y) := yb � 1 + ab6 (b2(x� 1) + ay2)2q(b) := b(1 � ap25)1� b :Observe that H2(x; y) = (h1(x; y); h2(x; y))and H�2(x; y) = (h�11 (x; y); h�12 (x; y)):Theorem 7.1. Assume the parameter values a, b > 0 lie in a su�ciently small openneighborhood of f(a; b) j a = 1:8; 0 < b � 0:025g and thateN := N � [0; q(b) + �]for some � > 0, but arbitrarily small. Then, there exists a semi-conjugacy from Inv( eN;H2)onto the full shift dynamics on two symbols, i.e. the invariant set of the horseshoe.The proof of this Theorem occupies the rest of this section and will be done in threestages each of which is meant to describe a method for applying the Conley index theoryto maps. As was indicated in the introduction, this result should be viewed as an example,included for the purposes of demonstrating the applicability of our abstract theorems. Itis not intended to be a careful analysis of the Henon map.The �rst stage of the proof involves a singular perturbation technique. It is well knownthat for b = 0, the invariant set of H is given by the invariant set of a 1-dimensional map.Using this map we �nd isolating neighborhoods and compute the Conley indices. We thenshow that the index information for the 1-dimensional map lifts to the two dimensionalmap which in turn implies that the index computations are valid for 0 � b � �. Thislifting of one dimensional information to two dimensions justi�es calling this a singularperturbation technique. 34

The second stage is referred to as a perturbation with a priori bounds; the estimatesof the one dimensional map are used to determine an interval of parameter values, a = 1:8,0 � b � 0:015 on which the index computations remain valid. Observe that this is asigni�cantly stronger result since we have a lower bound on the set of parameter values forwhich the theorem is valid. The importance of this with regards to modeling of physicalphenomenon is obvious.The third stage uses the continuation property of the index. The goal is to see forwhat parameter values the isolating neighborhood eN remains an isolating block. In ourcase this is shown to be hold over the interval a = 1:8, 0:015 � b � 0:025.7.A. Singular Perturbation.As was mentioned above we begin our proof by considering H2 with b = 0, i.e.H2(x; y) = (1 � a(1 + y � ax2)2; 0):This implies that Inv(N � [�y0; y0];H2) � R� f0gwhich allows us to restrict our attention to the one dimensional mapF (x) = 1� a(1 � ax2)2the graph of which is shown in Figure 3We shall now show that, aside from the fact that F is not a homeomorphism, allthe hypothesis of Theorem 2.3 are satis�ed. The dynamics of maps of the form of Fare well understood (see [De]) thus we leave the details of the following argument to thereader. Let N1 = [p1; p4] and N2 = [p5; p8]. A simple calculation shows that for a = 1:8,F (pi) 62 N1[N2 for i = 1; 4; 5; 8, and hence, N1[N2 is an isolating neighborhood. BecauseF is one dimensional and monotone over the sets N1 and N2, it is easy to check that wecan choose N11 = [p3; p4] N22 = [p7; p8]N12 = [p5; p6] N21 = [p1; p2] :Again the monotonicity of F over N1 and N2, makes the calculations thatCon1(Inv(Nk; F );Z2) � (Z2; id) k = 1; 2;35

and that �kl = 1 k; l = 1; 2;straightforward.Figure 3. Graph of F (x) = 1� 1:8(1� 1:8x2)2:Proposition 7.2. Given y0 > 0 and a = 1:8, there exists � > 0 such that for all b 2 [0; �],N � [�y0; y0] is an isolating neighborhood under H2. FurthermoreCon1(Inv(Nk � [�y0; y0];H2);Z2) � (Z2; id) k = 1; 2and �kl = 1 k; l = 1; 2;where the four smaller isolating neighborhoods are given byN11 = [p3; p4]� [�y0; y0] N22 = [p7; p8]� [�y0; y0]N12 = [p5; p6]� [�y0; y0] N21 = [p1; p2]� [�y0; y0]:Proof: N is an isolating neighborhood for F and for b = 0, Inv(N � [�y0; y0];H2) �R�f0g, therefore given y0 > 0, there exists � > 0 such that for all b 2 [0; �], N�[�y0; y0] isan isolating neighborhood under H2. This last comment is the standard local continuation36

property of the Conley index. Now, the commutativity property of the Conley index (see[Mr4]) comes into play to guarantee thatCon1(Inv(Nkl � [�y0; y0];H2);Z2) � Con1(Inv(Nkl; F );Z2) k; l = 1; 2:For the same reason, the �kl's remain the same.We can apply Theorem 2.3 for 0 < b < �. However, this only provides a semi-conjugacyof a power of H2 with the shift dynamics. In theorem 7.1 we claim that the power can bechosen to be 1. To see this one merely applies the proof of Theorem 2.4 at this point.Lemma 7.3. For a = 1:8 and 0 < b < �,Inv(N � [�y0; y0];H2) = Inv(N � [0; y0];H2)Proof: It is su�cient to observe that for x 2 [p1; p8] and jyj � y0 for y0 su�ciently small,1 + y � 1:8x2 > 0.7.B. Perturbation with A priori Bounds.In this case we shall present a fairly simple analytic technique for showing that The-orem 7.1 is valid on a �xed set of parameter values. The idea of the proof is intuitivelyquite simple: since our isolating neighborhood eN consists of rectangles, for b close to zerothe image of eN under H2 should be made up of attened rectangles whose bases are de-termined by F acting on N � f0g. What needs to be shown is that the invariant set in eNdoes not intersect the top, N � fq(b)g;bottom, N � f0g;or sides �i = pi � [0; q(b)] i = 1; : : : ; 8of the rectangles.This is done in the following three lemmas.37

Lemma 7.4. If x 2 N , a = 1:8 and b > 0, then(x; 0) 62 Inv( eN;H2):Proof: h2(x; y) > 0 for all y � 0.Lemma 7.5. If x 2 N , a = 1:8 and b > 0, then(x; q(b)) 62 Inv( eN;H2):Proof: We shall show that for all (x; y) such that x 2 N and 0 � y � q(b), h2(x; y) � q(b).h2(x; y) = b(1 + y � 1:8x2)� b(1 + q(b) � 1:8p25)� q(b)It is important to observe that the only points at which one obtains equality are (p4; q(b))and (p5; q(b)). In the next lemma it will be shown that for the prescribed values of a andb, h1(p4; q(b)); h1(p5; q(b)) 62 N .Lemma 7.6. For a = 1:8 and 0 < b � 0:015,h1(�i) \N = ;:Proof: De�ne G(x; y) := bx � ay2 � 2ay + 2a2yx2:Then h1(x; y) = F (x) +G(x; y). Let � = 0:05802. We leave it to the reader to check that� < minfjf(pk)� plj j k; l = 1; : : : ; 8g:We shall now show that for (x; y) 2 N � [0; q(b)],jG(x; y)j < �:Observe that since we know that the endpoints of N under F are mapped outside of N ,this �nishes the proof. 38

Our �rst claim is that the maxima and minima of G(x; y) restricted to eN occur onthe boundary. To see this observe that@G@y = 2a(�1� y + ax2):Hence, @G@y < 0 for (x; y) 2 eN . Therefore, the maximum of G on eN occurs at a point ofthe form (x; 0) and the minimum occurs at a point of the form (x; q(b)). Now G(x; 0) = bxhence the maximum occurs at (p8; 0). But for 0 < b < 0:01, jG(p8; 0)j < �. From@G@x (x; q(b)) it is easy to determine that the minimum value of G occurs at (p4; q(b)) andhence it is straightforward to check that jG(p4; q(b))j < � for the desired range of b.By continuation of the Conley index, the indexes of the various isolating neighborhoodsremain constant for all 0 < b � 0:015 and hence the results of Proposition 7.2 can beextended to b = 0:015. It is worth mentioning that jG(p4; q(0:025))j 6< �, thus we need adi�erent technique to complete the proof of Theorem 7.17.C. ContinuationThe results of 7.B. were obtained by using the estimates for b = 0. These estimatesare not su�cient in the range of values 0:015 � b � 0:025 which we shall now study. Insome sense, this is the region of parameter values at which the results become interesting,i.e. the estimates which determine the dynamics at these parameter values must be donein the full two dimensional system. Lemmas 7.4 and 7.5 are valid for all parameter values,however, we can no longer assume that h1(�i)\N = ;. Thus, we need to study the imagesof the sets �i, i = 1; : : : ; 8.The following observation is trivial but crucial. If (x; y) 2 Inv( eN ), then:p2i�1 � h1(x; y) � p2i i = 1; 2; 3; 4(32) p2i�1 � h�11 (x; y) � p2i i = 1; 2; 3; 4:(33) We shall make use of the following functions;g(x; p; b) = ax2 � 1 +r1 + bx � paf(x; p; b) = br bp+ 1� xa :39

Lemma 7.7. If (x; y) 2 Inv( eN ), then0 � y � g(x; p2i�1; b)g(x; p2i; b) � y i = 1; 2; 3; 4:sketch of proof: Beginning with the inequalities of (32) one solves for y in terms of xand p. Since we are only concerned with the points (x; y) which lie in eN , we ignore thosevalues of y which are less than zero. This leads directly to the function g.A similar proof, based on the inequalities (33), leads to the following lemma.Lemma 7.8. If (x; y) 2 Inv( eN ), theng(x; p2i�1; b) � y � g(x; p2i; b) i = 1; 2; 3; 4:As always our goal is to show that eN is an isolating neighborhood, i.e. that Inv( eN )\@ eN = ;. In fact, as will be seen for these parameter values eN is an isolating block. Thus,given Lemmas 7.4 and 7.5 it is su�cient to show that if (x; y) 2 �i for some i = 1; : : : ; 8,then H2(x; y) 62 eN .As a �rst step observe that @f@p > 0 and @g@p < 0. Also, for a = 1:8 and 0:015 � b � 0:025@f@b (pi; pj ; b) = 2(1� pi) + (bpj (2 + a)2q bpj+1�xa> 0while @g@b (pi; pj ; b) = pi2q1+bpi�pja :The important observation is that f and g are monotone in b. Furthermore, for pi, i =1; 2; 3; 4, they slope in opposite directions, while for pi, i = 5; 6; 7; 8, and 0:015 � b � 0:025,@f@b (pi; pj ; b) > @g@b (pi; pj ; b) (this last statement reduces to solving for the sign of a cubic in bover the interval in question). Therefore, to show that (32) and (33) are not simultaneouslysatis�ed for any point x 2 �i, i = 1; : : : 8, is reduced to evaluating f and g at the points40

(pi; pj ; 0:015) and (pi; pj ; 0:025). Doing this results in the following inequalities:f(p1; p8; b) < g(p1; p8; b)g(p2; p5; b) < f(p2; p1; b) < f(p2; p8; b) < g(p2; p4; b)g(p3; p5; b) < f(p3; p1; b) < f(p3; p8; b) < g(p3; p4; b)g(p4; p1; b) < f(p4; p1; b)g(p5; p1; b) < f(p5; p1; b)g(p6; p5; b) < f(p6; p1; b) < f(p6; p8; b) < g(p6; p4; b)g(p7; p5; b) < f(p7; p1; b) < f(p7; p8; b) < g(p7; p4; b)f(p8; p8; b) < g(p8; p8; b);and hence, eN is an isolating block.7.D. Concluding Remarks.The proof of Theorem 7.1 was essentially done by continuation. As was discussedin the introduction, this is typical of many applications of �xed point theory and moregenerally of most types of topological existence results. The continuation was done alongthe line of parameter values a = 1:8 and 0 � b � 0:025. However, the standard localcontinuation theory for the Conley index extends the results to an open neighborhood ofthis line.The same analysis can obviously be done for di�erent values of a = 1:8. However, asone decreases a one needs to take higher powers of H in order for the corresponding onedimensional map F to exhibit the horseshoe dynamics as explicitly as it occurs at a = 1:8.Similarly, the analysis can be extended to larger values of b. However, eN ceases to be anisolating block, and hence, to show that �i \ Inv( eN ) = ; one needs to take higher iteratesof hk and h�1k . For higher iterates, the inequality formulas which correspond to f and gbecome analytically intractable (the problem reduces to solving high order polynomials).Numerically, it is easy to check that eN remains an isolating neighborhood for values of bwell in excess of 0.1. Furthermore, using interval arithmetic one could undoubtably provesuch a result.We have chosen not to pursue these directions of extending Theorem 7.1 at this41

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