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Here systems (1) and (2) are differential inclusions. The details on the differential inclusions can be found in [3], for example. Thus, in the classification of systems without well-posedness, it is neces- sary to consider a type of “topological equivalence,” which does not com- pletely ignore the parametrization. There are several notions of “systems” that can be used as the objects of our equivalence. While set-valued flows are often used regarding the consideration of stability, it appears difficult to formulate the condition of orbit preservation in this setting [4, 14]. A more feasible candidate is the axiomatic theory of the solutions of ordi- nary differential equations. In particular, Filippov’s formalism and Yorke’s formalism have potential as the foundation of our methods, although neither considers the morphism between systems [9, 17]. In these theories, the solu- tion set of an ordinary differential equation is considered as a set of partial maps on R satisfying certain axioms. While Filippov’s formalism is fairly developed and applicable to a wide variety of problems, it is based on par- tial maps with closed domains and requires more axioms, which makes the generalization or the consideration of morphisms relatively difficult. Yorke’s formalism is based on partial maps with open domains; only a few axioms are required and it has been announced that basic results of the topological dynamics have been generalized. However, the proofs of these results have apparently not been published, as mentioned in [11]. In this article, we consider Yorke’s formalism in the setting of the action of topological groups, which can be seen as a generalization of topological dynamics [6, 7]. In particular, we define the notion of morphism and thereby obtain a notion of “topological equivalence” applicable to systems without well-posedness. During this process, we state and prove some generalization of the basic results announced in [17]. The remainder of this article is organized as follows. In Section 2, we define the space of partial maps and introduce Yorke’s formalism of topo- logical dynamics based on the star-construction. In Section 3, we consider the shift map of the partial maps and its relationship with the topological transformation groups and dynamical behavior. In Section 4, we introduce the notion of morphism of the star-construction and equivalences of systems without well-posedness, and discuss their relation with the usual definition of topological equivalence. In Section 5, we consider some examples to il- lustrate the formalism developed in this article. Finally, in Section 6, we provide concluding remarks.
2. Space of partial maps and Yorke’s formalism In this section, we introduce the basic concepts and constructions used throughout the article. In the following, X is a second-countable metric space and G is a locally compact second-countable Hausdorff topological group. Note that G is metrizable by the Birkhoff-Kakutani theorem. EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 3
Definition 2.1 (Partial maps). A continuous map φ : D X a partial map from G to X if D G is a nonempty open set. → ⊂ The set of all partial maps is denoted by Cp(G, X). For each φ : D X, we set dom φ := D. → A partial map φ C (G, X) with a nonempty connected domain is max- ∈ p imally defined if, for all ψ Cp(G, X) with a nonempty connected domain, the condition dom φ dom∈ψ and φ = ψ on dom φ implies φ = ψ. ⊂ Our aim is to topologize Cp(G, X) by introducing the compact-open topol- ogy. However, as the domains may vary, a slight modification is needed. The following construction is obtained from Abd-Allah and Brown [1]. Let ω / X and Xˆ := X ω . The open sets of Xˆ are defined by Xˆ and ∈ ∪ { } the open sets of X. Now we define a bijection µ : C (G, X) C(G, Xˆ ), p → where C(G, Xˆ ) is the set of all continuous maps from G to Xˆ topologized by the usual compact-open topology, by setting φ(x) (x dom φ) µ(φ)(x) := ∈ (ω (otherwise).
The topology on Cp(G, X) is defined so that it makes µ a homeomorphism. It has a subbasis given by the sets of the form W (K,U) := φ C (G, X) K dom φ, φ(K) U . { ∈ p | ⊂ ⊂ } The compact-open topology coincides with the topology of compact con- vergence. Lemma 2.2. The compact-open topology coincides with the topology of com- pact convergence. Namely, Cp(G, X) is a second countable space and φn φ in C (G, X) as n if and only if, for all compact subset K dom→φ, p → ∞ ⊂ K dom φn for sufficiently large n and supt K d(φn(t), φ(t)) 0 as n .⊂ ∈ → → ∞ Proof. The proof of the second countability of Cp(G, X) can be found in [2] (Proposition 2.1). Though the rest of the proof is similar to that of the classical one, we present it for the sake of completeness. Let φn φ in Cp(G, X) as n . Fix a compact subset K dom φ and a positive→ number ǫ> 0. For each→∞x K, we may find a relatively⊂ compact ∈ neighborhood Ux of x such that U dom φ, x ⊂ φ(U ) B (φ(x)), x ⊂ ǫ/3 φ(U¯ ) B (φ(x)). x ⊂ ǫ/2 n Because K is compact, there exist x1,x2, ,xn K with K i=1 Uxi . ¯ · · · ∈ n ⊂ Let Ki := Uxi K and Vi := Bǫ/2(φ(x)). Then we have φ i=1 W (Ki,Vi). n∩ ∈ S Therefore, i=1 W (Ki,Vi) is a neighborhood of φ in Cp(G, X) and there n T exists n0 N such that φn W (Ki,Vi) for all n n0. ∈ T ∈ i=1 ≥ T 4 TOMOHARU SUDA
Fix x K. Then x Ki for some i and we have for all n n0, ∈ ∈ ǫ ≥ ǫ d(φ(x), φ (x)) d(φ(x), φ(x )) + d(φ(x ), φ (x)) < + = ǫ. n ≤ i i n 2 2 Therefore, we conclude that supt K d(φn(t), φ(t)) ǫ. ∈ ≤ Conversely, let φn φ in the compact convergence and take a neighbor- n → hood i=1 W (Ki,Vi) of φ. For each i, there exists ri > 0 with B (φ(K )) V . T ri i ⊂ i Let r := min(ri). For each i, by the definition of compact convergence, there exists ni N with ∈ K dom φ , i ⊂ n sup d(φn(t), φ(t)) < r t Ki ∈ for all n ni. If we let n0 = max(ni), we have supt Ki d(φn(t), φ(t)) < r for all n n≥ and i = 1, 2, , n. Therefore, we deduce∈ for each t K , ≥ 0 · · · ∈ i φ (t) B (φ(t)) V . n ∈ r ⊂ i Thus, we obtain φ n W (K ,V ). n ∈ i=1 i i From the next lemma,T we may obtain maximally-defined partial maps. Lemma 2.3. Let φ C (G, X) with a nonempty connected domain. Then, ∈ p there exists a partial map φ¯ Cp(G, X) which is maximally defined and satisfies φ = φ¯ on dom φ. ∈
Proof. We define an order on the set Cc := φ Cp(G, X) dom φ is connected by setting φ ψ if dom φ dom ψ and φ({x)=∈ ψ(x) for all| x dom φ. This } ≤ ⊂ ∈ is clearly a partial order on Cc. Fix φ Cc and consider φ := ψ Cc φ ψ . As φ φ, φ is nonempty.∈ Let be a chain inC . We{ define∈ a| map≤ } by setting∈ C C S Cφ S dom := dom ψ W S ψ _ [∈S (x) := ψ(x) if x dom ψ. S ∈ Because is totally ordered,_ is well-defined, , and ψ S S S ∈ S ≤ S for all ψ . Therefore, from Zorn’s lemma, we see that φ has a maximal element,∈ which S is the desired partialW map. W C W Maximally defined partial maps have properties similar to those of the solution curves of an ordinary differential equation. The set of all maxi- mally defined partial maps is denoted by Cs(G, X). Clearly, C(G, X) with compact-open topology is a subspace of Cs(G, X). We define the orbit (φ) of φ C (G, X) by O ∈ s (φ) := φ(g) g dom(φ) . O { | ∈ } In general, the space Cs(G, X) is not Hausdorff and the limit of a sequence is not unique. However, we have the following property. EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 5
Lemma 2.4. Let φn n be a convergent sequence in C(G, X) with φn φ as n . If φ { ψ }as n in C (G, X), then φ = ψ. → →∞ n → →∞ s Consequently, if φni is a subsequence of φn n convergent in Cs(G, X), φ φ as i {in C}(G, X). { } ni → →∞ s Proof. By Lemma 2.2, we have φ = ψ on dom(ψ) G = dom(φ). Because ψ C (G, X), we have φ = ψ. ⊂ ∈ s Further, the next construction by Yorke enables us to avoid the difficulty accompanied with Cs(G, X) [17].
Definition 2.5 (Star-construction). For a subset S Cs(G, X) and W G X, we define ⊂ ⊂ × S∗W := (g, φ) φ S, g dom φ, (g, φ(g)) W . { | ∈ ∈ ∈ } Because S W G C (G, X), the topology of S W is naturally defined by ∗ ⊂ × s ∗ the subspace topology. We call S∗W the star-construction defined by S and W. Basic properties of the star-construction may be summarized as follows: Lemma 2.6. The following properties hold: (1) For S,S C (G, X) and W, W G X,S S and W W ′ ⊂ s ′ ⊂ × ⊂ ′ ⊂ ′ implies S∗W S′∗W ′ and S∗(W ′ W )= S∗W ′ S∗W. (2) For S C (G,⊂ X) and W G \X (µ M),\ we have ⊂ s µ ⊂ × ∈
S∗ Wµ = S∗Wµ, µ M µ M [∈ [∈ and
S∗ Wµ = S∗Wµ. µ M µ M \∈ \∈ (3) For subsets S C (G, X) (λ Λ) and W G X, we have λ ⊂ s ∈ ⊂ ×
( Sλ)∗W = Sλ∗W, λ Λ λ Λ [∈ [∈ and
( Sλ)∗W = Sλ∗W. λ Λ λ Λ \∈ \∈ Proof. The proofs are straightforward.
The next theorem enables us to analyze the “solution space” S∗W in a way similar to that of the usual ordinary differential equations.
Theorem 2.7. For all S Cs(G, X) and W G X, S∗W is second countable and Hausdorff. ⊂ ⊂ × 6 TOMOHARU SUDA
Proof. G C (G, X) is a second countable space by Lemma 2.2. Conse- × s quently, S∗W is also second countable. For the Hausdorff property, let (g, φ), (g , ψ) S W with (g, φ) = (g , ψ). ′ ∈ ∗ 6 ′ If g = g , we may find open neighborhoods U ,U ′ G of g and g with 6 ′ g g ⊂ ′ U U ′ = . Then we have U V U ′ V = for all open neighborhoods g ∩ g ∅ g × ∩ g × ′ ∅ V,V ′ of φ and ψ. If φ = ψ, the maximality of domains implies that dom φ dom ψ = or φ(g ) =6 ψ(g ) for some g dom φ dom ψ. In the former∩ case we have∅ 0 6 0 0 ∈ ∩ g = g′, which we have already considered. In the latter case, there exists r6 > 0 with B (φ(g )) B (ψ(g )) = . Because W ( g ,B (φ(g ))) and r 0 ∩ r 0 ∅ { 0} r 0 W ( g0 ,Br (ψ(g0))) are disjoint, we have U W ( g0 ,Br(φ(g0))) U ′ W ({ g } ,B (ψ(g ))) = for all neighborhoods×U and{ U} of g and g .∩ × { 0} r 0 ∅ ′ ′ Remark 2.8. The star construction may be characterized as follows. We establish the evaluation map ev : G Cs(G, X) G Xˆ by ev(g, φ) := (g,µ(φ)(g)), and the projection p : G × C (G, X) →C (×G, X) by p(g, φ) := × s → s φ. Note that these maps are continuous. Then the set S∗W is the largest subset E of G C (G, X) such that ev(E) W and p(E) S. In this sense, × s ⊂ ⊂ the set S∗W characterizes the “Cauchy problem” on W with the solution set S, which amounts to finding a map φ S satisfying φ(g) = x for each (g,x) W . ∈ ∈ In the next definition, we list the main additional axioms proposed by Yorke, which are abstractions of the conditions for well-posedness. Definition 2.9. Let S C (G, X). ⊂ s (1) The subset S satisfies the compactness axiom if S∗W is compact for all compact W G X. ⊂ × (2) The subset S satisfies the existence axiom on W if S∗ (t,x) is nonempty for all (t,x) W . { } ∈ (3) The subset S satisfies the uniqueness axiom on W if S∗ (t,x) is empty or a singleton for all (t,x) W . { } (4) The subset S has domain D if D ∈ dom φ for all φ S. ⊂ ∈ Remark 2.10. Because S∗W is second countable, S satisfies the com- pactness axiom if and only if S∗W is sequentially compact for all compact W G X. ⊂ × The next result, which is an analogue of the classical convergence theorem (Chapter 2, Theorem 3.2 in [10]), clarifies that the compactness axiom is an abstraction of the continuous dependence on the initial conditions.
Theorem 2.11. A subset S Cs(G, X) satisfies the compactness axiom if and only if the following property⊂ holds: if (g ,x ) (g,x) in G X and n n → × there exists a sequence of maps φn S with φn(gn) = xn, there is a map ψ S with ψ(g)= x and a subsequence∈ (g φ ) with (g , φ ) (g, ψ). ∈ { ni ni } ni ni → Proof. Let S Cp(G, X) satisfy the compactness axiom and (gn,xn) (g,x) be a sequence⊂ in G X with a corresponding sequence of maps φ →S × n ∈ EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 7 satisfying φ (g )= x . Then W := (g,x) (g ,x ) n N is compact, n n n { } ∪ { n n | ∈ } and therefore, S∗W is sequentially compact. Because (gn, φn) S∗W for each n, we may take a convergent subsequence (g , φ ) (g ,∈ ψ) S W. ni ni → ′ ∈ ∗ It is clear that g = g′. Conversely, let W G X compact and (g , φ ) (n = 1, 2, ) be a ⊂ × n n · · · sequence in S∗W. Then, (gn, φn(gn)) W for n = 1, 2, , and therefore, we can find a convergent subsequence∈ (g , φ (g )) · ·( ·g,y) W (i ni ni ni → ∈ → ). From the hypothesis, there exist a subsequence (gn , φn ) and a pair ∞ ij ij (g, ψ) S∗W satisfying (gn , φn ) (g, ψ) as j and ψ(g) = y. ∈ ij ij → → ∞ Therefore, S∗W is sequentially compact. From the classical convergence theorem, we immediately observe that the solution space of an ordinary differential equation satisfies the compactness axiom: Corollary 2.12. Consider an ordinary differential equation
x′ = f(t,x), where f : R W W is continuous and W Rn is a nonempty open set. Then the solution× → space S satisfies the compactness⊂ axiom. Further, a topological transformation group can be identified with a func- tion space satisfying the compactness axiom. Details on the topological transformation groups may be found in [5]. Corollary 2.13. Let π : G X X be a continuous left G-action on X. Then the set of maps × → S := π( ,x) x X { · | ∈ } satisfies the compactness, existence, and uniqueness axioms if X is locally compact. Proof. The existence and uniqueness axioms are satisfied because we have 1 S∗ (t,x) = t,π ,π(t− ,x) { } · for each (t,x) G X. For the compactness∈ × axiom, we show that π ,π(t 1,x ) π ,π(t 1,x) · n− n → · − as n whenever (t ,x ) (t,x) as n . Let y := π(t 1,x ) and n n n n− n y := →π(t ∞1,x), and fix a compact→ subset K→ ∞ G. We may find a neigh- − ⊂ borhood U of y with U¯ being compact and yn U for sufficiently large n. Because π is continuous, π is uniformly continuous∈ on K U.¯ Therefore, we × have sups K d(π(s,yn),π(s,y)) 0 as n . ∈ → →∞ In some cases, the compactness axiom is not independent with other ax- ioms and certain restrictions are present.
Corollary 2.14. Let subset S Cs(G, X) satisfy the compactness axiom and there exists a subset W ⊂G X such that S satisfies the existence axiom on W . If X is locally compact,⊂ × then S satisfies the existence axiom on the closure of W . 8 TOMOHARU SUDA
The following theorem is a generalization of a well-known result on the global solution.
Theorem 2.15. Let S Cs(G, X) satisfy the compactness axiom and φ S. If there is a compact⊂ set K with (φ) K, dom φ coincides with∈ a connected component of G. O ⊂ Proof. There is a connected component G of G satisfying dom φ G . c ⊂ c If dom φ = Gc, ∂ (dom φ) = . Then we may take a sequence gn g ∂ (dom φ)6 with g dom φ.6 Because∅ the image of φ is contained in→K, we∈ n ∈ may assume φ(gn) y K. By Theorem 2.11, there exists (g, ψ) with (g , φ) (g, ψ). Therefore,→ ∈ g dom φ, which is a contradiction. ni → ∈ The next theorem is a generalization of a result announced by Yorke.
Theorem 2.16. If S Cs(G, X) satisfies the compactness axiom and X is locally compact, S W ⊂is metrizable for each subset W G X. ∗ ⊂ × Proof. First we show that S∗W is locally compact. Fix (g, φ) S∗W. Be- cause G and X are locally compact, we may take relatively compact∈ neigh- borhoods U and V of g and φ(g) such that φ(U¯) V . Then S (U¯ V¯ ) S W ⊂ ∗ × ∩ ∗ is a compact neighborhood of (g, φ) in S∗W. This is verified by observing
(g, φ) U W (U,V¯ ) S∗W S∗(U¯ V¯ ) S∗W. ∈ × ∩ ⊂ × ∩ Because S∗W is locally compact and Hausdorff, it is regular. Using the second countability of S∗W (Theorem 2.7), we may apply the Urysohn metrization theorem.
3. Shift of partial maps and dynamical behavior Based on the concepts introduced in the previous section, we consider generalizations of the usual notions in dynamical systems theory. The concept of shift invariance plays a central role in the discussion in this section. First we establish that the shift map is a G-action.
Theorem 3.1 (Shift map). The shift map σ : G Cp(G, X) Cp(G, X), which is defined by × → σ(g, φ)(x) := φ(xg) for x dom(φ)g 1, is continuous and satisfies the following conditions: ∈ − (1) For each φ Cp(G, X) we have σ(e, φ)= φ. (2) For all g, h ∈ G and φ C (G, X), we have σ(g,σ(h, φ)) = σ(gh, φ). ∈ ∈ p That is, σ is a left G-action. 1 Proof. To show that σ is continuous, it suffices to confirm that σ− (W (K,V )) 1 is open for each compact K G and open V X. Let (g, φ) σ− (W (K,V )) . This is equivalent to ⊂ ⊂ ∈ σ(g, φ)(K)= φ (Kg) V. ⊂ EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 9
Therefore, we have φ W (Kg, V ) . We may find an open neighborhood U of g such that U¯ is compact∈ and
1 KU¯ dom(φ) φ− (V ), ⊂ ∩ using the regularity of G. Then we have
1 (g, φ) U W KU,V¯ σ− (W (K,V )) . ∈ × ⊂ 1 Thus, σ− (W (K,V )) is open. The other assertions are proven via direct calculations.
Remark 3.2. By the definition of the shift map, we have
1 dom (σ(g, φ)) = dom (φ) g− for each (g, φ) G Cp(G, X). Further, the following identity holds for all left G-action π∈: G × X X : × → (2) σ (g,π( ,x)) = π ( ,π(g,x)) . · · Thus, the triplet (G, σ, Cp(G, X)) is a topological transformation group. As the name “shift system” usually refers to discrete systems, here we call it the Bebutov system on Cp(G, X). Now we clarify the relationship between the Bebutov system and the usual topological transformation groups. This is given by the next theorem, which is a generalization of a result by Yorke (Theorem 2.3 in [17]). Theorem 3.3. Let X be locally compact. Then a σ-invariant subset S ⊂ Cs(G, X) satisfies the compactness, existence, and uniqueness axioms and has domain G if and only if it is given by a left G-action πS : G X X on X via × → (3) S := π ( ,x) x X . { S · | ∈ } Proof. The sufficiency is a consequence of Corollary 2.13 and the identity (2). Let a σ-invariant subset S C (G, X) satisfy the compactness, existence, ⊂ s and uniqueness axioms and have domain G. We establish a map πS : G X X by × → πS(g,x) := φ(e,x)(g), where φ(e,x) is the unique element in S with φ(e,x)(e) = x. We note that σ g, φ S and (e,x) ∈
σ g, φ(e,x) (e)= φ(e,x)(g)= πS(g,x). Therefore, we have
(4) φ(e,πS(g,x)) = σ g, φ(e,x) . 10 TOMOHARU SUDA
From this identity, we obtain
πS (g,πS (h, x)) = φ(e,πS(h,x))(g)
= σ h, φ(e,x) (g)
= σ g,σ h, φ(e,x) (e) = σ gh, φ (e,x) (e) = φ( e,x)(gh) = πS(gh,x).
Finally, we show that πS is continuous. Let (gn,xn) (g,x) as n in G X. From Theorem 2.11 and the uniqueness, we have→ φ →φ ∞ as × (e,xn) → (e,x) n in C (G, X). By Theorem 3.1, we obtain →∞ s σ g , φ σ g, φ n (e,xn) → (e,x) in C (G, X). Evaluation at e gives us π (g ,x ) π (g,x) in X. s S n n → S Corollary 3.4. Let X be locally compact. If a σ-invariant subset S ⊂ Cs(G, X) satisfies the compactness, existence, and uniqueness axioms and has domain G, S is homeomorphic to X. Furthermore, (S,σ) and (X,πS ) are isomorphic as topological transformation groups. Proof. Let us consider the map p : S X defined by → p(φ)= φ(e) and s : X S defined by → s(x)= φ(e,x). The continuity of p follows from Lemma 2.2 and that of s from the identity s(x)= πS( ,x). Clearly, p s = idX and s p = idS. The identity· (4) can be◦ rewritten as ◦
s(πS(g,x)) = σ (g,s(x)) . Further, the following identity holds:
p (σ (g, φ)) = πS(g,p(φ)).
Hence, (S,σ) and (X,πS ) are isomorphic as topological transformation groups. Thus, each topological transformation groups on X can be interpreted as a σ-invariant subset of Cs(G, X), and its asymptotic behavior can be stated in terms of the functional space Cs(G, X). Notions regarding the dynamical behavior can be generalized to a σ- invariant subset of Cs(G, X). First we note the following property of the “initial value problem,” which simplifies the definitions. Lemma 3.5. Let S be a σ-invariant subset of C (G, X) and x X. If s ∈ x = φ(g) for some φ S and g dom(φ), there exists φˆ S with φˆ(e)= x. ∈ ∈ ∈ Proof. Set φˆ = σ(g, φ) S. ∈ EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 11
Definition 3.6 (Weak invariance, equilibrium point). For a σ-invariant subset S of Cs(G, X), we define the following: (1) A subset A X is weakly invariant with respect to S if, for each x A, there⊂ exists (e, φ) S (e, x) with (φ) A. ∈ ∈ ∗{ } O ⊂ (2) A point x X is an equilibrium of S if there exists (e, φ) S∗ (e, x) with (φ)=∈ x . ∈ { } O { } The following are some generalizations of well-known properties of these notions.
Lemma 3.7. Let S be a σ-invariant subset of Cs(G, X). For each φ S, (φ) is weakly invariant. ∈ O Proof. Let x (φ). By definition, there exists g dom(φ) with φ(g)= x. ∈ O ∈ Then we have (e, σ(g, φ)) S∗ (e, x) . Further, (σ(g, φ)) = (φ). Hence, (φ) is weakly invariant. ∈ { } O O O Lemma 3.8. Let S be a σ-invariant subset of Cs(G, X) satisfying the com- pactness axiom and X be locally compact. If A X is compact and weakly invariant with respect to S, A¯ is also weakly invariant.⊂
Proof. Let x A.¯ Then we may take a sequence xn A with xn x as n . Because∈ A is weakly invariant, we may{ take,}⊂ for each n, φ→ S → ∞ n ∈ with φn(e) = xn and (φn) A. Let U be a relatively compact open neighborhood of x. ForO sufficiently⊂ large n, we have
(e, φ ) S∗( e U¯ ). n ∈ { }× By the compactness of S, we may assume that there exists (e, φ) S ( e ∈ ∗ { }× U¯ ) with (e, φn) (e, φ) as n . Therefore, φ(e) = limn φn(e) = x. Further, for each→g dom(φ), we→ ∞have →∞ ∈
φ(g) = lim φn(g). n →∞ Because φ (g) A, φ(g) A.¯ n ∈ ∈ Note that an intersection of weakly invariant sets need not be weakly invariant due to the lack of uniqueness.
4. Morphisms of the star-construction and topological equivalence As mentioned in Remark 2.8, a star-construction can be characterized in terms of two maps ev : G C (G, X) G X and p : G C (G, X) × s → × × s → Cs(G, X). Morphisms of the star-construction can be defined so that they respect this structure. 12 TOMOHARU SUDA
Definition 4.1 (Morphisms of the star-construction). Let S Cs(G, X),S′ C (G , X ), W G X and W G X . A morphism between⊂ the star-⊂ s ′ ′ ⊂ × ′ ⊂ ′ × ′ constructions S∗W and (S′)∗W ′ is a triplet of continuous maps
H : S∗W (S′)∗W ′ → k : W W ′ → η : S S′ → such that ev H = k ev, ◦ ◦ p H = η p. ◦ ◦ We denote a morphism by
Lemma 4.3. Let< H,k,η >: S∗W (S′)∗W ′ be a morphism. Then we have →
H (A∗B) η(A)∗k(B) (S′)∗W ′ ⊂ ⊂ for each A S and B W. In particular, if
Proof. By Lemma 2.6, A∗B S∗W. H (A∗B) η(A)∗k(B) is an immediate consequence of the definition⊂ of morphisms and⊂ Remark 2.8.
The axioms listed in Definition 2.9 are preserved by isomorphisms. Theorem 4.4. Let S C (G, X) and S C (G , X ). If S (G X) and ⊂ s ′ ⊂ s ′ ′ ∗ × (S′)∗(G′ X′) are isomorphic by < H,k,η >, the following assertions are true. ×
(1) The subset S satisfies the compactness axiom if and only if S′ does so. (2) The subset S satisfies the existence axiom on W if and only if S′ does so on k(W ). (3) The subset S satisfies the uniqueness axiom on W if and only if S′ does so on k(W ). EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 13
Proof. From Lemma 4.3, we have
H (S∗W )= η(S)∗k(W )
= (S′)∗k(W ). Because H is a homeomorphism, we obtain the results.
The equivalence class of subsets of Cs(G, X) under the isomorphism rela- tion is rather large. This classification can be regarded as that of the types of problems, as observed by the following result. Theorem 4.5. Let X be locally compact. If a σ-invariant subset S ⊂ Cs(G, X) satisfies the compactness, existence, and uniqueness axioms and has domain G, S (G X) is isomorphic to S (G X), where ∗ × 0∗ × S := ψ C (G, X) ψ (g)= x for all g G . 0 { x ∈ s | x ∈ } Proof. By Theorem 3.3, there exists a continuous map πS : G X X such that × → S = π ( ,x) x X . { S · | ∈ } We establish a morphism
Definition 4.6 (Phase space-preserving morphism). Let S Cs(G, X),S′ C (G , X ), W G X and W G X . A morphism⊂< H,k,η > be-⊂ s ′ ′ ⊂ × ′ ⊂ ′ × ′ tween the star-constructions S∗W and (S′)∗W ′ preserves phase space if k has the form k = (τ, h), where τ : W G′ and h : W X′ are continuous and h(g,x)= h(g ,x) for all (g,x), (g →,x) W. → ′ ′ ∈ S∗W and (S′)∗W ′ are isomorphic via phase space-preserving isomor- phisms if there exists a phase space-preserving isomorphism < H,k,η > 1 1 1 between S∗W and (S′)∗W ′ such that
If a morphism preserves phase space and W is open, there is a map induced on the “phase space.” Lemma 4.8. Let S C (G, X), S C (G , X ), W G X and W ⊂ s ′ ⊂ s ′ ′ ⊂ × ′ ⊂ G′ X′. Let Wˆ := x X (g,x) W for some g G . If a morphism < H,k,η× > preserves{ phase∈ | space and∈ W is open in∈G } X, there exists × a continuous map hˆ : Wˆ X′ defined by setting hˆ(x) := h(g,x) for some g W := g G (g,x) →W . ∈ x { ∈ | ∈ } Proof. Let the canonical projection p : W Wˆ be defined by p (g,x) := X → X x. It suffices to show that pX is an identification map because we have the following commutative diagram (for details, see Ch 4, Theorem 3.1 in [8]): W
❅ h pX ❅ ❄ ❅❅❘ ✲ Wˆ X′ hˆ Let Uˆ Wˆ be open in Wˆ . Then there exists an open subset U X such ˆ⊂ ˆ 1 ˆ 1 ˆ ⊂ that U = U W . Because pX− (U) = (G U) W, pX− (U) is open in W. ∩ ˆ ˆ 1 ˆ × ∩ Conversely, let U W and pX− (U) be open in W. Then there exists an ⊂ 1 ˆ ˆ open subset V G X such that pX− (U) = V W. For each x U, we may find g G⊂with× (g,x) V W. Because V∩ W is open, there∈ exist open subsets∈U G and U ∈ X∩with (g,x) U ∩ U V W. 1 ⊂ 2 ⊂ ∈ 1 × 2 ⊂ ∩ Now we show that U2 Wˆ U.ˆ For each x′ U2 Wˆ , we have (g,x′) ∩ ⊂ ∈ ∩ 1 ∈ U U for all g U . Therefore, (g,x ) V W = p− (Uˆ). By definition, 1 × 2 ∈ 1 ′ ∈ ∩ X x = p (g,x ) U.ˆ ′ X ′ ∈ Remark 4.9. Because k(g,x) = (τ(g,x), hˆ(x)) W holds for some g ∈ ′ ∈ G for each x Wˆ , we have hˆ : Wˆ Wˆ ′ := x′ X′ (g′,x′) W for some g ∈ G . → { ∈ | ∈ ′ ′ ∈ ′} Remark 4.10. For a phase space-preserving isomorphism < H,k,η > be- 1 1 1 tween S∗W and (S′)∗W ′, < H− , k− , η− > also preserves phase space if and only if hˆ is injective. The property in the following lemma is crucial in the discussion below. Lemma 4.11. Let S C (G, X), S C (G , X ), W G X and ⊂ s ′ ⊂ s ′ ′ ⊂ × W ′ G′ X′. If a morphism < H,k,η > preserves phase space, then we have⊂ × η(φ)(τ(g,x)) = hˆ(x), for each (g, φ) S (g,x) , where k = (τ, h). ∈ ∗{ } Proof. Let (g, φ) S (g,x) . From p H = η p, we have ∈ ∗{ } ◦ ◦ H(g, φ) = (t′, η(φ)) EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 15 for some t G . Because ev H = k ev, we obtain ′ ∈ ′ ◦ ◦ (t′, η(φ)(t′)) = k(g, φ(g)) = (τ(g, φ(g)), h(g, φ(g))). Therefore, η(φ)(τ(g,x)) = hˆ(x).
Theorem 4.12. Let the star-constructions S∗W and (S′)∗W ′ be isomorphic via phase space-preserving isomorphisms, where W and W ′ are open. Then, Wˆ and Wˆ are homeomorphic. Further, W is homeomorphic to W for ′ x hˆ′(x) each x Wˆ . ∈ Proof. Let the canonical projection pX : W Wˆ be defined by pX (g,x) := → ˆ x. Similarly, we define the canonical projection pX′ : W ′ W ′. If we set 1 → k− = (˜τ, h˜), we have the following commutative diagram, and it is imme- diately clear that hˆ : Wˆ Wˆ ′ is a homeomorphism: → − k ✲ k 1✲ W W ′ W
❅ h ❅ ˜ p ′ h p X ❅ pX ❅ X ❄ ❅❅❘ ❄ ❅❅❘ ❄ ✲ ✲ Wˆ Wˆ ′ Wˆ hˆ h˜ˆ Now we show that W is homeomorphic to W for each x Wˆ . We define x hˆ′(x) ∈ τ : W W by τ (g) := τ(g,x). This is well-defined and continuous x x → hˆ′(x) x because k(g,x) = (τ(g,x), hˆ(x)) W ′ ∈ for each g Wx. To show that τx is a homeomorphism, it suffices to confirm ∈ ˆ ′ that k(Wx x ) = Wˆ′ h(x) because it implies τx τ˜hˆ(x) = idW × { } h(x) × { } ◦ hˆ(x) andτ ˜ τ = id . By definition, we have hˆ(x) ◦ x Wx k(W x ) W ′ hˆ(x) x × { } ⊂ hˆ(x) × { } 1 k− (W ′ hˆ(x) ) W x . hˆ(x) × { } ⊂ x × { } Therefore, k(W x )= W hˆ(x) . x × { } hˆ′(x) × { } The notion of isomorphism via phase space-preserving isomorphisms re- spects basic dynamical properties.
Theorem 4.13. Let the star-constructions S∗(G X) and (S′)∗(G′ X′) be isomorphic via phase space-preserving isomorphisms.× Then we have× hˆ ( (φ)) = (η(φ)) O O for all φ S. ∈ Proof. Let φ S. If y hˆ ( (φ)) , we may find g dom(φ) such that ∈ ∈ O ∈ y = hˆ(φ(g)) by definition. From Lemma 4.11, we have η(φ)(τ(g, φ(g))) = hˆ(φ(g)). 16 TOMOHARU SUDA
Therefore, y (η(φ)) . Conversely,∈ let O y (η(φ)) . By definition, there exists g dom (η(φ)) ∈ O ′ ∈ such that y = η(φ)(g′). Therefore, (g′, η(φ)) (S′)∗ (g′,y) . Let (g0, φ) := 1 ∈ { } H− (g′, η(φ)). Then we have τ(g0, φ(g0)) = g′, and
y = η(φ)(g′)= η(φ)(τ(g0, φ(g0))) = hˆ(φ(g0)). Therefore, y hˆ ( (φ)) . ∈ O Corollary 4.14. Let the star-constructions S (G X) and (S ) (G X ) ∗ × ′ ∗ ′ × ′ be isomorphic via phase space-preserving isomorphisms, where S and S′ are σ-invariant. Then A X is weakly invariant if and only if hˆ(A) is weakly ⊂ invariant. In particular, x0 X is an equilibrium if and only if hˆ(x0) is an equilibrium. ∈ Proof. Because we may consider the inverse of the morphism, it suffices to show that A is weakly invariant if hˆ(A) is so. Let x A. Because hˆ(x) hˆ(A) and hˆ(A) is weakly invariant, we may ∈ ∈ find ψ S such that ψ(e)= hˆ(x) and (ψ) hˆ(A). Therefore, we have ∈ ′ O ⊂ 1 hˆ (η− (ψ)) = (ψ) hˆ(A) O O ⊂ 1 by Theorem 4.13. Because hˆ is a homeomorphism, (η− (ψ)) A. O ⊂1 By Lemma 3.5, the proof is complete if we show x (η− (ψ)). This follows from ∈ O 1 hˆ(x)= ψ(e) (ψ)= hˆ (η− (ψ)) . ∈ O O Therefore, A is weakly invariant. From Theorem 4.13, it is clear that x X is an equilibrium if and only 0 ∈ if hˆ(x0) is an equilibrium. Not only are the orbits preserved, but their parametrization is also in good correspondence.
Theorem 4.15. Let the star-constructions S∗(G X) and (S′)∗(G′ X′) be isomorphic via phase space-preserving isomorphisms.× For each φ ×S, we ∈ define a map φ : dom(φ) dom(η(φ)) by φ(g) := τ(g, φ(g)). Then, φ is a homeomorphism.D → D D Further, if S and S have domains G and G , the map : S G ′ ′ D × → G′ defined by (φ, g) := φ(g) is continuous. In particular, if S is path connected, thenD all maps inD the family : G G φ S are isotopic. {Dφ → ′ | ∈ } To prove this theorem, we observe the following property of the map . Dφ Lemma 4.16. Let the star-constructions S (G X), (S ) (G X ) and ∗ × ′ ∗ ′ × ′ (S′′)∗(G′′ X′′) be isomorphic via phase space-preserving isomorphisms < H,k,η >:×S (G X) (S ) (G X ) and
′′(g)= τ ′ τ(g, φ(g)), hˆ(φ(g)) Dφ = τ ′ (τ(g, φ(g)), η(φ)(τ(g, φ(g))))
= ′ (g), Dη(φ) ◦ Dφ for each g dom(φ). ∈ Proof of Theorem 4.15. From Lemma 4.11, φ is well-defined and its con- tinuity is obvious. To show that it is a homeomorphism,D it suffices to show that ( ) 1 = , but this follows immediately from Lemma 4.16. Dφ − Dη′ (φ) If S and S′ have domains G and G′, the continuity of follows from that of the evaluation map because we have (φ, g)= τ(g, φ(gD)) = τ(ev(g, φ)). D An example of a path connected set S Cs(G, X) is given by the following lemma. ⊂ Lemma 4.17. Let Φ : G X X be a G-action and X be path connected and locally compact. Then,× S =→ Φ( ,x) x X is path connected. { · | ∈ } Proof. Let φ0, φ1 S, where φi = Φ( ,xi) for i = 0, 1. Because X is path connected, there exists∈ a continuous· map γ : [0, 1] X. Then, Γ(s) := Φ( , γ(s)) is the desired path between φ and φ . → · 0 1 A change of variables yields a phase space-preserving isomorphism.
Theorem 4.18. Let S Cs(G, X) and h : X X′ and τ : G G′ be homeomorphisms. If we⊂ set → → 1 S′ := h φ τ − φ S C (G, X), { ◦ ◦ | ∈ }⊂ s then S∗(G X) and (S′)∗(G′ X′) are isomorphic via phase space-preserving isomorphisms.× × Proof. We define a morphism
Considering Theorem 4.15, we may define a generalization of the concept of topological equivalence and topological conjugacy. 18 TOMOHARU SUDA
Definition 4.19 (Topological equivalence). Let S C (G, X), S C (G, X ), ⊂ s ′ ⊂ s ′ where S and S′ have the domain G. S and S′ are topologically equivalent if S (G X) and (S ) (G X ) are isomorphic via phase space-preserving ∗ × ′ ∗ × ′ isomorphisms and these morphisms can be taken so that φ is isotopic to the identity of G and (e)= e for all φ S. D Dφ ∈ S and S′ are topologically conjugate if S∗(G X) and (S′)∗(G X′) are isomorphic via phase space-preserving isomorphisms× and these morphisms× can be taken so that is the identity on G for all φ S. Dφ ∈ Lemma 4.20. Topological equivalence is an equivalence relation. Proof. This is a consequence of Remark 4.7, Lemma 4.16, and the property of isotopy class. If G = R and S is given by a flow, these definitions relate to the usual one as follows. Lemma 4.21. A continuous function f : R R is isotopic to identity if and only if it is monotonically increasing and→ bijective. Proof. If f : R R is isotopic to identity, it is clear from the definition that f is monotonically→ increasing and bijective. Conversely, if f is monotonically increasing and bijective, the map H : [0, 1] R R defined by × → H(s,x) := sf(x) + (1 s)x − is an isotopy between f and the identity. Lemma 4.22. Let f : X Y Z be a continuous map. If f(x, ) : Y Z is a homeomorphism for× each→x X, there exists a continuous· map→g : X Z Y satisfying ∈ × → f(x, g(x, z)) = z and g(x,f(x,y)) = y for all x X, y Y and z Z. ∈ ∈ ∈ Proof. First we show that f is an open map. Let W X Y be open and take z f(W ). By definition, we may find (x,y) X⊂ Y ×with f(x,y)= z. Let U ∈ X and V Y be open neighborhoods∈ of× x and y such that U V ⊂W. Then we⊂ have × ⊂ z f(U V )= f(u, V ) f(W ). ∈ × ⊂ u U [∈ Because f(u, ) is a homeomorphism for each u U, f(U V ) is open. Therefore, f is· an open map. ∈ × Now we define f¯(x,y) := (x,f(x,y)). Then it is open, continuous and bijective. Therefore, f¯ is a homeomorphism. Therefore, we may find a continuous map g : X Z Y satisfying × → f(x, g(x, z)) = z EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 19 and g(x,f(x,y)) = y for all x X, y Y and z Z. ∈ ∈ ∈ Theorem 4.23. Let the star-constructions S∗(R X) and (S′)∗(R X′) be described by flows, that is, there exist flows Φ× : R X X and× Φ : R X X such that × → × ′ → ′ S = Φ( ,x) x X { · | ∈ } S′ = Ψ( ,x′) x′ X′ . { · | ∈ } Assume that X and X′ are locally compact. Then S and S′ are topologically equivalent if and only if there exist a homeomorphism h : X X′ and a continuous map τ : R X R such that → × → Ψ(τ(t,x), h(x)) = h(Φ(t,x)) for all (t,x) R X, where τ( ,x) is monotonically increasing and bijective, and τ(0,x) =∈ 0 for× all x X.· ∈ Also, S and S′ are topologically conjugate if and only if there exists a homeomorphism h : X X such that → ′ Ψ(t, h(x)) = h(Φ(t,x)) for all (t,x) R X. ∈ × Proof. Let S and S′ be topologically equivalent. Let a map : S R R be that in Theorem 4.15. We define D × → τ¯(t,x) := (Φ( ,x),t). D · From Lemma 4.11, we have η(Φ( ,x))(¯τ (t,x)) = hˆ(Φ(t,x)). · By considering the case t = 0, we deduce η(Φ( ,x)) = Ψ( , hˆ(x)), · · which implies Ψ(¯τ(t,x), hˆ(x)) = hˆ(Φ(t,x)). Because (Φ( ,x), ) is isotopic to the identity,τ ¯(t,x) is monotonically increasingD and· bijective· for all x X by Lemma 4.21. Also, we have ∈ τ¯(0,x) = x for all x X by definition. The map hˆ : X X′ is a homeo- morphism by Theorem∈ 4.12. → Conversely, let there exist a homeomorphism h : X X′ and a continu- ous map τ : R X R such that → × → Ψ(τ(t,x), h(x)) = h(Φ(t,x)) for all (t,x) R X, where τ( ,x) is monotonically increasing and bijective, and τ(0,x) =∈ 0 for× all x X. · ∈ 20 TOMOHARU SUDA
We define a morphism
Therefore, Φ( ,x)(t)= τ(t,x). If (Φ( ,x)) has a self-intersection, Ψ( , h(x)) is periodic.D Let· T be the period.O Then· we have · τ( t, Φ(t,x)) + τ(t,x)= n(t)T − for some n(t) Z. Because n(t) = (τ( t, Φ(t,x))+τ(t,x))/T is a continuous function that∈ takes integer values, it− is a constant function. By considering t = 0, we have n(t) = 0. Therefore, we have Φ( ,x)(t) = τ(t,x) also in this case. D · The proof for the case of topological conjugacy is similar. Remark 4.24. Thus, our definition of topological equivalence is stronger than the usual one. Indeed, in the usual definition, the continuity of τ is not guaranteed. However, there are examples where τ can be retaken so that it becomes continuous. It is not clear whether such regularization is always possible.
5. Applications In this section, we apply the formalism given in the preceding sections to some examples to illustrate its use and properties. Example 1. The solution sets of the following systems are not isomorphic. (1)x ˙ [1/2, 1]. (2)x ˙ ∈ 1/2, 1 . (3)x ˙ =∈ { 1. } EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 21
Indeed, (1) and (3) satisfy the compactness axiom, while inclusion (2) does not. The uniqueness axiom is satisfied only in equation (3). From Theorem 4.4, these axioms are invariant under isomorphisms. Therefore, these three examples are not isomorphic. Proof. It is an immediate consequence of the Arzel`a-Ascoli theorem that the solution set of inclusion (1) satisfies the compactness axiom. For inclusion (2), consider a sequence φ of solutions defined inductively by { n} 1 1 t + 4 (t< 2 ) 1 1 − 1 φ0(t)= t ( t< ) 2 − 2 ≤ 2 t 1 ( 1 t) − 4 2 ≤ and 1 3 2 φn(2t + 1) 4 (t< 0) φn+1(t)= − 1 φ (2t 1) + 3 (0 t). ( 2 n − 4 ≤ In C(R, R), φ converges to ψ given by { n} t + 1 (t< 1) 4 − ψ(t)= 3 t ( 1 t< 1) 4 − ≤ t 1 (1 t), − 4 ≤ which is not a solution of inclusion (2). By Lemma 2.4, it follows that no subsequence of φn converges to a solution of inclusion (2) in Cs(R, R). From Theorem 2.11,{ } this implies that the solution set of inclusion (2) does not satisfy the compactness axiom. It is clear that the uniqueness axiom is satisfied only in equation (3). Example 2. Let us consider the following family of equations with a pa- rameter a R : ∈ 2 (5) x′ = x + a. R2 R2 Let the corresponding solution set be Sa. Then Sa∗ and Sb∗ are isomor- phic via a phase space-preserving isomorphism if a and b have the same sign. Proof. The solution φa of (5) satisfying the initial condition φa (t )= (t0,x0) (t0,x0) 0 x0 is given by 1 √a tan √a(t t0) + tan− (x0/√a) (a> 0) − 1 √ a tanh √ a(t t0) + tanh− ( x0/√ a) (a< 0, x0 < √ a) φa (t)= − − − − − − | | − (t0,x0) √ √ a (a< 0,x0 = a) ± − √ a ± − − − −1 (a< 0, x0 > √ a) tanh(√ a(t t0)+tanh ( √ a/x0)) | | − − − − − From these expressions, we have a 1 1 x0 π 1 1 x0 π dom(φ )= t0 tan− ,t0 tan− + (t0,x0) − √a √a − 2√a − √a √a 2√a 22 TOMOHARU SUDA for a> 0 and
R ( x0 √ a) a 1 1 | |≤ − dom(φ )= ( ,t0 tanh− ( √ a/x0)) (x0 > √ a) (t0,x0) √ a −∞ − − − − − 1 1 √ √ (t0 √ a tanh− ( a/x0), ) (x0 < a) − − − − ∞ − for a< 0. In either case, the maps a b η(φ(t ,x )) := φ 0 0 (√a/bt0,√b/a x0) a b k(t,x) := t, x b a r r ! a H(t, φ)= t, η(φ) b r R2 R2 give a phase space-preserving isomorphism between Sa∗ and Sb∗ . It can be verified directly that the domains are also preserved, as in Theorem 4.15. Example 3. The solution sets of the following systems are topologically conjugate. (1)x ˙ [1/2, 1]. (2)x ˙ ∈ [1, 2]. (3)x ˙ ∈ [ 1, 1/2]. ∈ − − Proof. Let the solution set of (1) be S1, that of (2) be S2, and that of (3) R2 R2 be S3. We define an isomorphism between S1∗ and S2∗ by
η1(φ) := 2φ
k1(t,x) := (t, 2x)
H1(t, φ) := (t, η1(φ)).
Because φ(t)= t for all φ S1, S1 and S2 are topologically conjugate. DR2 R2 ∈ For S1∗ and S3∗ , we define η (φ) := φ 2 − k (t,x) := (t, x) 2 − H2(t, φ) := (t, η2(φ)).
Then S1 and S3 are topologically conjugate. Example 4. For a compact metric space X, let us consider the group of automorphisms Aut(X) and the space C(X) of all continuous functions on X, each topologized by the compact-open topology. We define SX := C(Aut(X),C(X)) and W := Aut(X) C(X). X × If X and Y are homeomorphic, SX∗ WX and SY∗ WY are isomorphic via a phase space-preserving isomorphism. EQUIVALENCE OF TOPOLOGICAL DYNAMICS WITHOUT WELL-POSEDNESS 23
Proof. Let h : X Y be a homeomorphism. Then we have a natural isomorphism τ : Aut(→X) Aut(Y ) defined by τ(g) := h g h 1 and the → ◦ ◦ − pullback h∗ : C(Y ) C(X), which is a homeomorphism. We define the maps→ 1 1 η(φ) := (h∗)− φ τ − ◦ ◦ 1 k(g,f) := (τ(g), (h∗)− (f)) H(g, φ) = (τ(g), η(φ)) , for each g Aut(X), f C(X) and φ SX . Using the continuity of compositions,∈ it can be shown∈ that these three∈ maps are homeomorphisms. Thus, SX∗ WX and SY∗ WY are isomorphic via a phase space-preserving iso- morphism.
6. Concluding remarks As the purpose of this article is to present a basic framework, further study on its application to various problems is necessary. In particular, bifurcation or structural stability can be studied once we have introduced a proper notion of the “distance” between systems. Regarding the dynamical behavior, our discussion here has been limited to weakly invariant sets and equilibrium points. However, it appears possible to generalize other notions too.
Acknowledgements This study was supported by a Grant-in-Aid for JSPS Fellows (20J01101).
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Faculty of Mathematics, Keio University Email address: [email protected]