ON CONLEY’S FUNDAMENTAL THEOREM OF DYNAMICAL SYSTEMS

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ON CONLEY’S FUNDAMENTAL THEOREM OF DYNAMICAL SYSTEMS

M. R. RAZVAN

Received 25 February 2002 To the memory of Charles C. Conley

We generalize Conley’s fundamental theorem of dynamical systems in Conley index theory.

We also conclude the existence of a regular index filtration for every Morse decomposition.

2000 Mathematics Subject Classification: 37B25, 37B30, 37B35.

1. Introduction. Conley is mostly known for his fundamental theorem of dynami- cal systems and his homotopy index theory [1]. In the former, he proved that every continuous flow on a compact metric space admits a Lyapunov function which strictly decreases along nonchain recurrent orbits. This result has been developed by Franks for homeomorphisms [3] and Hurley for noncompact metric spaces [5,6,7,8]. In the latter, Conley defined a homotopy invariant for any isolated invariant set for a contin- uous flow. This invariant gives some valuable information about the behavior of the isolated invariant set. This paper concerns a combination of these two masterpieces.

Indeed, we show the existence of Conley’s Lyapunov function on every index pair in the sense of Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition.

2. Conley index theory. Letϕ^t be a continuous flow on a metric spaceX. Aniso- lated invariant set is a subsetS⊂Xwhich is the maximal invariant set in a compact neighborhood of itself. Such a neighborhood is called an isolating neighborhood. A Morse decompositionforS is a collection{Mi}ⁿi=1, where eachMiis an isolated invari- ant subset ofS and for allx∈S−n

i=1Mi, there existi, j∈ {1, . . . , n}such thati > j, α(x)∈Mi, andω(x)∈Mj. A pair(A, A^∗)of subsets ofSis called anattractor-repeller pair if{A, A^∗}is a Morse decomposition forS, that is,α(x)∈A^∗ andω(x)∈Afor everyx∈S−(A∪A^∗).

Let S be an isolated invariant set with an isolating neighborhood V and a Morse decomposition{Mi}ⁿ_i=1. In [11], it is proved that ifϕ^[0,+∞)(x)⊂V, thenω(x)⊂Mifor some 1≤i≤n. Similarly, ifϕ^(−∞,0](x)⊂V, thenα(x)⊂Mifor some 1≤i≤n. Now, forj=0, . . . , n, we define

I_j⁺=I_j⁺(V )=

x∈V|ϕ^[0,∞)(x)⊂V , ω(x)⊂M_j+1∪···∪Mn

, I_j⁻=I_j⁻(V )=

x∈V|ϕ^(−∞,0](x)⊂V , α(x)⊂M1∪···∪Mj

, S_j^∗=

x∈S|ω(x)⊂Mj+1∪···∪Mn , Sj=

x∈S|α(x)⊂M1∪···∪Mj

.

(2.1)

Moreover, if(A, A^∗)is an attractor-repeller pair forS, we set

I_A⁺∗=

x∈V|ϕ^[0,∞)(x)⊂V , ω(x)⊂A^∗ , I_A⁻=

x∈V|ϕ^(−∞,0](x)⊂V , α(x)⊂A

. (2.2)

In [2,11], it is proved thatI_j⁺andI_j⁻are compact and(Sj, S_j^∗)is an attractor-repeller pair forS. This fact allows us to prove our results for an attractor-repeller pair and then extend them to every Morse decomposition.

In order to define the concept of index pair, we follow [9,11]. Given a compact pair (N, L)withL⊂N⊂X, we define the induced semiflow onN/Lby

ϕ^t_#:N/L →N/L, ϕ^t_#(x)=





ϕ^t(x) ifϕ^[0,t](x)⊂N−L,

[L] otherwise. (2.3)

In [9], it is proved thatϕ^t_#is continuous if and only if (i) Lis positively invariant relative toN, that is,

x∈L, t≥0, ϕ^[0,t](x)⊂N ⇒ϕ^[0,t](x)⊂L, (2.4)

(ii) every orbit which exitsNgoes throughLfirst, that is,

x∈N, ϕ^[0,∞)(x)⊂N ⇒ ∃t≥0, ϕ^[0,t](x)⊂N, ϕ^t(x)∈L, (2.5)

or equivalently ifx∈N−L, then there is at >0 such thatϕ^[0,t](x)⊂N.

Definition2.1. Anindex pair for an isolated invariant setS⊂Xis a compact pair (N, L)in X such thatN−Lis an isolating neighborhood for S and the semiflowϕ_#^t induced byϕ^t is continuous.

In [1,2,9,11], it has been shown that every isolated invariant setS admits an index pair (N, L)and the homotopy type of the pointed spaceN/L is independent of the choice of the index pair. TheConley indexofSis the homotopy type of(N/L, [L]).

Note2.2. We will not distinguish betweenN−LandN/L−{[L]}.

Definition2.3. An index pair(N, L)is calledregular if the exit time map defined by

τ₊:N →[0,+∞], τ₊(x)=



 sup

t|ϕ^[0,t](x)⊂N−L

ifx∈N−L,

0 ifx∈L, (2.6)

is continuous.

Proposition2.4. An index pair(N, L)is regular provided thatϕ^[0,t](x)⊂N−Lfor everyx∈Landt >0.

The above result provides a criterion for the regularity of index pairs. The reader is referred to [11] for the details about regular index pairs and the proof of this useful criterion.

Definition 2.5. Let S be an isolated invariant set with a Morse decomposition {Mi}ⁿi=1. An index filtration is a sequence N0⊂N1⊂ ··· ⊂Nn of closed subsets of Xsuch that(Nk, Nk−1)is an index pair forMkfor every 1≤k≤nand(Nn, N0)is an index pair forS. When each(Nk, N_k−1)is regular, then the above filtration is called a regular index filtration.

It is well known that every Morse decomposition admits an index filtration [11]. We desire to show that every Morse decomposition admits a regular index filtration.

3. Conley’s fundamental theorem. In this section, we construct a Lyapunov function on an index pair by modifying Conley’s original argument [1,10]. The key point is that one should work withI_A⁺∗and[L]∪I_A⁻instead of the attractor-repeller pair(A, A^∗). The following lemma is the main idea in the proof of the continuity of Conley’s Lyapunov function.

Lemma3.1. LetSbe an isolated invariant set with an attractor-repeller pair(A, A^∗), an index pair(N, L), and the isolating neighborhoodV=N−L. IfBis a compact subset ofN/L−I_A⁺∗ andU is a neighborhood of[L]∪I_A⁻, then there exists T ∈R⁺ such that ϕ^{[T ,}_# ^+∞)(B)⊂U.

Proof. We may assume thatU is a compact neighborhood of[L]∪I_A⁻ with U∩ I_A⁺∗ = . Now, suppose that there are xn ∈B and tn → ∞ such thatϕ_#^tn(xn)∈ U.

Since B is compact, we may choose xn’s in N−L so that xn → x ∈ B. It is easy to see that ϕ^[0,+∞)(x)⊂N−L. Since B∩I⁺_A∗ =, we have ω(x)⊂A, hence there is a t∈R⁺ such that ϕ^[t,+∞)(x)∈U. Since^◦ xn→x, there aret_n∈[t, tn] such that t_n−t→ ∞,ϕ^[t,tn)(xn)⊂U, and^◦ ϕ^tn(xn)∈∂Ufor every sufficiently largen∈N. There- fore, the sequenceϕ^tn(xn)has a limit pointy∈∂Usuch thatϕ^(−∞,0](y)⊂U∩(N−L) and y ∈ω(B)⊂S. Thus,α(y)⊂A, which means that y ∈I⁻(A). This contradicts y∈∂U.

Theorem3.2. LetSbe an isolated invariant set with an attractor-repeller pair(A, A^∗) and an index pair(N, L). There exists a continuous functiong:N/L→[0,1]such that

(i) g⁻¹(0)=[L]∪I_A⁻andg⁻¹(1)=I_A⁺∗,

(ii) g(ϕ_#^t(x)) < g(x)for everyx∈[L]∪I_A⁻∪I⁺_A∗andt∈R⁺.

Proof. Letρ:N/L→[0,1] be a continuous function withρ⁻¹(0)=[L]∪I_A⁻ and ρ⁻¹(1)=I_A⁺∗. We define h:N/L →[0,1] by h(x)=sup_t≥0ρ(ϕ^t_#(x)). It is not hard to see thath⁻¹(0)=[L]∪I_A⁻,h⁻¹(1)=I_A⁺∗. We show thathis upper-semicontinuous.

For everyx∈N/Land >0, there is at∈R⁺such thatρ(ϕ^t_#(x)) > h(x)−. Now, there is a neighborhoodUsuch thatρ(ϕ_#^t(y)) > h(x)−for everyy∈U. Therefore, h(y) > h(x)−for everyy ∈U, which proves the upper-semicontinuity ofh. As a result,his continuous inh⁻¹(1). Now, suppose thatx∈h⁻¹(1)=I_A⁺∗and <1−h(x).

If we setB=ρ⁻¹[0, h(x)+]andU=ρ⁻¹[0, h(x)+)in the above lemma, we obtain

aT∈R⁺withρ(ϕ_#^t(y)) < h(x)+for everyy∈Bandt≥T. Now, the by continuity of ϕ#, there exists an open setV⊂N/Lsuch thatx∈Vandρ(ϕ^t_#(y)) < h(x)+for every t∈[0, T ]andy∈V. Therefore,h(y) < h(x)−for everyy∈U∩V, which shows that his lower-semicontinuous inx. Now, it is left to check thatg:=_+∞

0 e^−th(ϕ^t_#(x))dtis the desired function.

Theorem3.3. LetSbe an isolated invariant set with an index pair(N, L)and a Morse decomposition{Mi}ⁿ_i=1. There is a continuous functiong:N/L→[0, n+1]such that

(i) g⁻¹(0)=[L]andg(Mi)=ifor every1≤i≤n, (ii) ifx∈N−L−n

i=1Miandt >0, theng(ϕ^t_#(x)) < g(x).

Proof. Consider the attractor-repeller pairs(Sj, S_j^∗)for 0≤j≤n. ByTheorem 3.2, there are continuous functionsgi:N/L→[0,1]with g⁻_i¹(0)=[L]∪I_j⁻, g_i⁻¹(1)=I_j⁺, andgj(ϕ^t(x)) < gj(x)for everyx∈[L]∪I_j⁻∪I_j⁺. Now,g:=g0+···+gnis the desired function.

Corollary3.4. Every Morse decomposition admits a regular index filtration.

Proof. Letf be the above Lyapunov function. If we setNk:=π⁻¹(f⁻¹[0, k+1/2]) for 0≤k≤nandNn:=N, then, byProposition 2.4,(Nk, N_k−1)is a regular index pair forMk, for every 1≤k≤n.

Definition3.5. Letϕ^t be a continuous flow on a compact metric spaceX. An- chain forϕ^tis a sequencex0, . . . , xninXandt1, . . . , tnin[1,+∞)such thatd(ϕ^ti(xi−1), xi) < . A pointx∈X is called chain-recurrent if for every >0, there is an-chain withx0=xn=x. The set of all chain recurrent points forϕ^tis denoted byR(ϕ^t).

It is not hard to check thatR(ϕ^t)is a closed invariant subset ofX containing the nonwandering setΩ(ϕ^t). In [1,10], it has been shown thatR(ϕ^t|R(ϕ^t))=R(ϕ^t)and R(ϕ^t)=

(A∪A^∗), where the intersection is taken over all attractor-repeller pairs (A, A^∗)inX. It is also known that the number of all attractor-repeller pairs in a compact metric space is at most countable.

Theorem3.6. LetSbe an isolated invariant set with an index pair(N, L). Then there is a continuous functiong:N/L→[0,1]such that

(i) g⁻¹(0)=[L]andg(ϕ^t_#(x))≤g(x)for everyx∈N/Landt≥0, (ii) ifx∈N−L−R(ϕ^t|S)andt≥0, theng(ϕ^t_#(x)) < g(x).

Proof. Let{(Ai, A^∗_i)}^∞i=1be the sequence of all attractor-repeller pairs inSincluding (, S)and(S, ). Now, byTheorem 3.2, there are continuous functionsgi:N/L→[0,1]

such thatg_i⁻¹(0)=[L]∪I_A⁻_i,g_i⁻¹=I⁺_A∗

i, andgi(ϕ^t_#(x)) < gi(x)for everyt∈R⁺ and x∈[L]∪I_A⁺∗

i ∪I_A⁻

i. Now,g= ^∞i=12⁻ⁱgiis the desired function.

The above result can be considered as a generalization of Conley’s fundamental the- orem of dynamical systems. A similar result for discrete dynamical systems can be obtained by following [4,12].

Acknowledgment. The author would like to thank the Institute for Studies in The- oretical Physics and Mathematics (IPM) for supporting this research.

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M. R. Razvan: Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395- 5746, Tehran, Iran

E-mail address:[email protected]